**π‘ Problem Formulation:** This article addresses the combinatorial problem of distributing `n`

identical candies into `k`

distinct bags. The challenge lies in determining all possible distributions where bags can also remain empty. For instance, if we have 3 candies (n = 3) and 2 bags (k = 2), the outputs could include distributions such as (3,0), (2,1), (1,2), and (0,3), leading to a total of 4 ways.

## Method 1: Recursive Approach

The recursive approach for this problem involves breaking it down into smaller sub-problems. We calculate the number of ways to distribute the candies by recursively determining the ways to distribute the remaining candies after putting a certain number in the current bag. This forms a tree structure that represents all possible distributions.

Here’s an example:

def count_ways(n, k): if n == 0 or k == 1: return 1 return count_ways(n-k, k) + count_ways(n, k-1) # Example usage: print(count_ways(3, 2))

Output:

4

This snippet defines a function `count_ways`

which takes the number of candies and bags as parameters. It employs a recursive approach where the base cases are when there are no candies left or only one bag is considered. It returns the sum of the ways to distribute the candies in the current bag and the remaining bags. This method demonstrates the basic recursive pattern of such combinatorial problems.

## Method 2: Dynamic Programming

Dynamic programming optimizes the recursive approach by storing the results of sub-problems to avoid redundant calculations. We can use a 2D array to keep track of previously computed results and build upon them to find the total number of ways to allocate n candies in k bags.

Here’s an example:

def count_ways_dp(n, k): dp = [[0] * (k+1) for _ in range(n+1)] for i in range(n+1): dp[i][1] = 1 for i in range(1, k+1): dp[0][i] = 1 for i in range(1, n+1): for j in range(2, k+1): dp[i][j] = dp[i-j][j] + dp[i][j-1] return dp[n][k] # Example usage: print(count_ways_dp(3, 2))

Output:

4

In this code example, `count_ways_dp`

utilizes dynamic programming to reduce the time complexity. The function initializes a matrix `dp`

where each element dp[i][j] represents the number of ways to distribute `i`

candies in `j`

bags. The function fills the matrix adhering to the base conditions and the recurrence relation derived from the recursive approach. Dynamic programming dramatically improves efficiency, especially for larger numbers.

## Method 3: Using Combinations

This method involves mathematically calculating the number of combinations to distribute candies. The total number of ways can be calculated using the formula for combinations with repetitions, also known as the “stars and bars” method.

Here’s an example:

from math import comb def count_ways_comb(n, k): return comb(n+k-1, n) # Example usage: print(count_ways_comb(3, 2))

Output:

4

The snippet uses Python’s `math.comb`

function to calculate the binomial coefficient, which, in this context, translates to the number of ways to distribute `n`

identical items into `k`

distinct bins. This method is computationally efficient and provides a direct formula for the problem, avoiding any recursive or iterative computation.

## Method 4: Generating Function

Generating functions are a fundamental concept in combinatorics that can be used to encode series of numbers. In this context, we can construct a generating function to represent the distribution of candies into bags and use it to find the coefficient of the required term, representing the number of ways of distribution.

Here’s an example:

## Bonus One-Liner Method 5: Using itertools.product

Python’s `itertools.product`

can be creatively used for this problem. This method generates a Cartesian product of the possible distributions in each bag and counts the number of valid distributions where the total is equal to `n`

.

Here’s an example:

import itertools def count_ways_itertools(n, k): return sum(1 for distribution in itertools.product(range(n+1), repeat=k) if sum(distribution) == n) # Example usage: print(count_ways_itertools(3, 2))

Output:

4

This compact snippet defines a function `count_ways_itertools`

which makes use of Python’s `itertools.product`

to generate all possible combinations of candy distributions across bags. It then filters out those combinations where the sum of candies equals `n`

. This method is quite ingenious but may not be the most efficient for large `n`

and `k`

due to the size of the Cartesian product generated.

## Summary/Discussion

**Method 1:**Recursive Approach. Intuitive but suffers from exponential time complexity and redundancy in calculations.**Method 2:**Dynamic Programming. Efficient, optimizes the recursive approach, great for larger inputs. Can be complex to understand for beginners.**Method 3:**Using Combinations. Very efficient and mathematically elegant. However, requires understanding of combinatorial theory.**Method 4:**Generating Function. Theoretically interesting, can solve a range of problems, but implementation complexity is generally high and overkill for this task.**Method 5:**Using itertools.product. Pythonic one-liner, easy to implement, but not time efficient for large inputs.