Distribute Candies Among Children II

Problem Description

You are given two positive integers n and limit. Return the total number of ways to distribute n candies among 3 children such that no child gets more than limit candies.

Key Insights

• Each child can receive between 0 and limit candies.
• The problem can be viewed as finding the number of non-negative integer solutions to the equation x1 + x2 + x3 = n, with constraints 0 <= xi <= limit.
• The generating functions or combinatorial counting methods can be applied to solve this efficiently.
• If n is greater than 3 * limit, then the number of distributions can be calculated using combinations.

Space and Time Complexity

Time Complexity: O(1)
Space Complexity: O(1)

Solution

To solve the problem, we can use combinatorial mathematics to calculate the number of ways to distribute candies. The main idea is to count the total distributions without constraints and then subtract the cases where one or more children exceed the limit.

1. Calculate the total number of ways to distribute n candies without any constraints.
2. Use the principle of inclusion-exclusion to subtract the cases where a child receives more than limit candies.
3. For each child, consider how many ways we can distribute the remaining candies after giving one child limit + 1 candies.

The approach utilizes combinatorial counting, specifically the "stars and bars" theorem, to enumerate distributions.

Code Solutions