# 5 Best Ways to Check for Triplets in Python Whose Sum Is Less than a Target Value

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π‘ Problem Formulation: This article tackles the challenge of finding the number of triplets in an array where the sum of each triplet is less than a given target value. For instance, given input array [2, 1, 4, 3, 6] and a target sum 8, a desirable output might be 4, representing the number of triplets that meet the criteria (e.g., [2, 1, 3], [1, 4, 2], [2, 3, 1], and [1, 3, 2]).

## Method 1: Brute Force Approach

This method involves checking all possible triplets in the array to see if their sum is less than the target value. It has a function specification that iterates over every combination of three elements and verifies the sum condition. This approach is straightforward but has a high time complexity of O(n^3).

Here’s an example:

```def countTripletsBruteForce(arr, target):
count = 0
n = len(arr)
for i in range(n):
for j in range(i + 1, n):
for k in range(j + 1, n):
if arr[i] + arr[j] + arr[k] < target:
count += 1
return count

print(countTripletsBruteForce([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.

Output:

`4`

This snippet defines a function `countTripletsBruteForce()` that accepts an array and a target sum. It utilizes three nested loops to generate each triplet and checks if their summed value is less than the target before incrementing the count. The final count is then returned.

## Method 2: Sorting and Two Pointers

By first sorting the array, we can use a two-pointer technique to reduce the time complexity. For a given first element, we find pairs in the remaining array whose sum with the first element is less than the target. The function specification implements this strategy, bringing the complexity down to O(n^2).

Here’s an example:

```def countTripletsTwoPointers(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
left, right = i + 1, len(arr) - 1
while left < right:
if arr[i] + arr[left] + arr[right] < target:
count += right - left
left += 1
else:
right -= 1
return count

print(countTripletsTwoPointers([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsTwoPointers()` function starts by sorting the array. It then iterates through the array, using two pointers to scan from both ends to find pairs that form a valid triplet with the current element. The count is incremented by the number of valid pairs found.

## Method 3: Using Binary Search

Binary search can be used after sorting the array to find the third element of the triplet that makes the sum less than the target value. This method is efficient for a large array with a complexity of O(n^2 log n).

Here’s an example:

```import bisect

def countTripletsBinarySearch(arr, target):
arr.sort()
count = 0
for i in range(len(arr) - 2):
for j in range(i + 1, len(arr) - 1):
t = target - arr[i] - arr[j]
count += bisect.bisect_left(arr, t, j + 1) - (j + 1)
return count

print(countTripletsBinarySearch([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsBinarySearch()` function uses the Python module `bisect` to efficiently find the insertion point where the third element of the triplet would be placed, without exceeding the target sum. The difference between this index and the current second element index gives the number of valid triplets for a particular pair.

## Method 4: Hashing

Hashing can be used to speed up the search for the third element in each potential triplet, resulting in a time complexity improvement in some cases. The trade-off is increased space complexity.

Here’s an example:

```def countTripletsHashing(arr, target):
count = 0
for i in range(len(arr)):
for j in range(i + 1, len(arr)):
if target - arr[i] - arr[j] in set(arr[j+1:]):
count += 1
return count

print(countTripletsHashing([2, 1, 4, 3, 6], 8))
```

Output:

`4`

In this `countTripletsHashing()` function, a set is used to check if the complement of the sum of two elements that would make the sum less than the target exists further in the array. This is done for each possible pair of elements in the array.

## Bonus One-Liner Method 5: List Comprehension with itertools

The itertools module provides a clean, Pythonic way to generate and filter triplets using a one-liner. Although elegant, this method usually has the same performance as the brute force approach.

Here’s an example:

```import itertools

def countTripletsItertools(arr, target):
return sum(sum(triplet) < target for triplet in itertools.combinations(arr, 3))

print(countTripletsItertools([2, 1, 4, 3, 6], 8))
```

Output:

`4`

The `countTripletsItertools()` function utilizes the `itertools.combinations()` method to generate all possible triplets and then uses a generator expression to count how many of these have a sum less than the target.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple to implement, but slow, with O(n^3) time complexity. Not suitable for large datasets.
• Method 2: Sorting and Two Pointers. More efficient than brute force, with O(n^2) time complexity. Requires sorting but is significantly faster for large arrays.
• Method 3: Using Binary Search. Combined with sorting, this method has a complexity of O(n^2 log n). It is more efficient than brute force and works well for substantially large datasets.
• Method 4: Hashing. Can offer performance improvements but has increased space complexity. The actual performance gain depends on dataset characteristics.
• Bonus Method 5: List Comprehension with itertools. Provides clean and Pythonic code. Performance is usually on par with the brute force method, but its simplicity makes it a one-liner solution.