#2928

Easy

LeetCode Problem Workspace

Distribute Candies Among Children I

Given two integers n and limit, find the number of ways to distribute n candies among 3 children, with no child receiving more than limit candies.

math combinatorics enumeration

Problem Statement

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

For example, given n = 5 and limit = 2, the valid distributions of candies are (1, 2, 2), (2, 1, 2), and (2, 2, 1), making the output 3.

Examples

Example 1

Input: n = 5, limit = 2

Output: 3

There are 3 ways to distribute 5 candies such that no child gets more than 2 candies: (1, 2, 2), (2, 1, 2) and (2, 2, 1).

Example 2

Input: n = 3, limit = 3

Output: 10

There are 10 ways to distribute 3 candies such that no child gets more than 3 candies: (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0) and (3, 0, 0).

Constraints

1 <= n <= 50
1 <= limit <= 50

Solution Approach

Brute-Force Approach

A straightforward method is to use three nested loops to check all possible ways of distributing candies. For each combination of candy counts, check if the sum equals n and if no child receives more than the limit.

Optimized Combinatorial Approach

An optimized approach would use combinatorics to count the number of ways to assign candies without exceeding the limit for each child. This avoids the need for nested loops and can provide a more efficient solution.

Dynamic Programming Approach

By using dynamic programming, you can store results of previously computed subproblems and build up the solution iteratively, reducing redundant calculations and improving time complexity.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The brute-force approach has a time complexity of O(limit^3), since we need to check each possible distribution. A combinatorial solution could reduce this to O(limit), depending on the method used, while dynamic programming could further optimize it, potentially reducing space complexity as well.

What Interviewers Usually Probe

Candidate suggests an efficient combinatorial approach.
Candidate considers dynamic programming for optimization.
Candidate does not consider the brute-force method but instead tries an optimized solution.

Common Pitfalls or Variants

Common pitfalls

Forgetting to check that the sum of candies matches n.
Incorrectly counting distributions where children receive more than the limit.
Overcomplicating the problem with unnecessary optimizations before checking the brute-force solution.

Follow-up variants

Varying the number of children to test scalability.
Adjusting the limit per child to explore edge cases.
Considering the possibility of using more advanced combinatorics techniques to optimize the solution.

FAQ

What is the best approach for solving the Distribute Candies Among Children I problem?

The brute-force approach is easy to implement but may not be efficient. A more optimal approach involves using combinatorics or dynamic programming to count valid distributions without unnecessary computations.

How do I handle the constraint of no child receiving more than the limit?

Ensure that the distribution checks never exceed the limit value for any child. This can be done in the brute-force solution by filtering out invalid distributions during iteration or through careful combinatorial counting.

What is the time complexity of the brute-force solution?

The brute-force solution has a time complexity of O(limit^3) because we are checking all possible distributions of candies using nested loops.

How can I optimize the solution to the Distribute Candies Among Children I problem?

You can optimize the solution by using combinatorics to calculate the number of valid distributions without generating each possible combination or by employing dynamic programming to avoid redundant calculations.

Does this problem have any edge cases I should be aware of?

Yes, edge cases include when n is very small compared to the limit, or when n is very large, testing the algorithm's ability to handle the upper bounds of the problem constraints.

terminal

Solution

Solution 1: Combinatorial Mathematics + Principle of Inclusion-Exclusion

According to the problem description, we need to distribute $n$ candies to $3$ children, with each child receiving between $[0, limit]$ candies.

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class Solution:
    def distributeCandies(self, n: int, limit: int) -> int:
        if n > 3 * limit:
            return 0
        ans = comb(n + 2, 2)
        if n > limit:
            ans -= 3 * comb(n - limit + 1, 2)
        if n - 2 >= 2 * limit:
            ans += 3 * comb(n - 2 * limit, 2)
        return ans

Continue Practicing

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