**π‘ Problem Formulation:** Finding all combinations in a list that satisfy a particular condition is a common task in programming, useful in fields such as data analysis and algorithm development. Suppose we have a list of integers and we want to find all unique combinations of a certain length where the sum is divisible by a given divisor. For example, given a list `[1, 2, 3, 4]`

, we want to find all pairs of numbers whose sum is divisible by 3, yielding the output `[(1, 2), (2, 4)]`

.

## Method 1: Using itertools.combinations

The itertools module provides a method called `combinations()`

that can be used to find all combination tuples of a given length from a list. To apply a condition, we can use a list comprehension along with the combinations.

Here’s an example:

import itertools def find_combinations(lst, length, divisor): return [combo for combo in itertools.combinations(lst, length) if sum(combo) % divisor == 0] example_list = [1, 2, 3, 4] result = find_combinations(example_list, 2, 3) print(result)

Output:

[(1, 2), (2, 4)]

This code defines a function that generates all combinations and filters them based on our condition. The resulting combination tuples form a list where the sum of numbers is divisible by the provided divisor.

## Method 2: Using a Nested Loop

A nested loop approach allows us to manually iterate over the list items and form combinations. This approach gives us the flexibility to apply any condition we want inside the loops without needing extra libraries.

Here’s an example:

def find_combinations(lst, length, divisor): combinations = [] for i in range(len(lst)): for j in range(i+1, len(lst)): if (lst[i] + lst[j]) % divisor == 0 and length == 2: combinations.append((lst[i], lst[j])) return combinations example_list = [1, 2, 3, 4] result = find_combinations(example_list, 2, 3) print(result)

Output:

[(1, 2), (2, 4)]

This code snippet manually iterates through the list and checks each pair for our condition, adding it to the results if the condition passes. It is a straightforward method but becomes less efficient with larger combinations.

## Method 3: Using Recursive Backtracking

Recursive backtracking is useful for exploring all potential combinations and allows for custom conditions to be easily integrated. This method is especially powerful in terms of understanding the underlying mechanism of combination generation.

Here’s an example:

def find_combinations(lst, length, divisor, start=0, out=None): if out is None: out = [] if length == 0: if sum(out) % divisor == 0: print(tuple(out)) else: for i in range(start, len(lst)): find_combinations(lst, length - 1, divisor, i + 1, out + [lst[i]]) example_list = [1, 2, 3, 4] find_combinations(example_list, 2, 3)

Output:

(1, 2) (2, 4)

The function recursively builds combinations and prints them only if they meet the condition. This is particularly efficient for generating combinations of varying lengths and imposing complex conditions.

## Method 4: Using Filter with itertools.combinations

Combining Python’s built-in `filter()`

function with `itertools.combinations()`

is a functional programming approach that promotes clean and readable code. This method applies the condition as a filter directly on the combination iterator.

Here’s an example:

import itertools def condition(combo, divisor): return sum(combo) % divisor == 0 example_list = [1, 2, 3, 4] combinations = itertools.combinations(example_list, 2) filtered_combinations = filter(lambda c: condition(c, 3), combinations) print(list(filtered_combinations))

Output:

[(1, 2), (2, 4)]

This snippet creates an iterator of combinations, then filters those that do not satisfy the condition and converts the result to a list. It’s both concise and functional.

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

A Python list comprehension can achieve the combination and condition check all in one concise expression. However, this is limited to combinations of a fixed, small length due to the hardcoded nature of comprehension.

Here’s an example:

example_list = [1, 2, 3, 4] print([(x, y) for x in example_list for y in example_list if x < y and (x + y) % 3 == 0])

Output:

[(1, 2), (2, 4)]

This one-liner creates a list of tuples directly, encompassing two loops to form pairs. It’s the most concise method but lacks scalability for longer combinations and readability for complex conditions.

## Summary/Discussion

**Method 1: itertools.combinations.**Powerful and readable. Ideal for generating combinations of any length. May require additional memory and runtime for generating all combinations before filtering.**Method 2: Nested Loop.**Simple to understand. No additional libraries required. Efficiency decreases with larger lists and combination lengths.**Method 3: Recursive Backtracking.**Offers deep control over the combination building process. Can be overkill for simple conditions or may become difficult to understand for complex scenarios.**Method 4: Filter with itertools.combinations.**Functional approach. Makes the code clean and declarative. Can be less intuitive for those unfamiliar with functional programming concepts.**Bonus Method 5: List Comprehension with Conditions.**Super concise. Best for small-scale problems and fixed-length combinations. Not easily extendable to more complex conditions or variable lengths.