#2929

Medium

LeetCode Problem Workspace

Distribute Candies Among Children II

Determine how to distribute n candies among 3 children without exceeding a limit on individual candies.

Problem Statement

You are given two integers: n (the number of candies) and limit (the maximum number of candies a child can receive). The task is to find the total number of ways to distribute n candies among 3 children such that no child gets more than the limit number of candies.

For example, if n = 5 and limit = 2, there are 3 ways to distribute the candies: (1, 2, 2), (2, 1, 2), and (2, 2, 1). The problem tests your ability to apply combinatorics to count valid distributions.

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 <= 106
1 <= limit <= 106

Solution Approach

Use Combinatorics to Count Valid Distributions

The problem boils down to counting the possible ways to distribute n candies to 3 children, each having a maximum of 'limit' candies. We can iterate over the possible candy distributions for each child, keeping track of how many valid combinations exist.

Enumerate Candy Distributions

You can enumerate the number of candies each child gets, ensuring the total sum equals n, and no child exceeds the limit. This process requires calculating the number of valid combinations through a combinatorial approach, factoring in constraints on the distribution.

Optimize for Efficient Calculation

Since the number of combinations grows quickly, it's important to find a way to calculate the result efficiently. The solution leverages mathematical formulas and dynamic programming to avoid brute-force enumeration, optimizing the time complexity.

Complexity Analysis

Metric	Value
Time	O(1)
Space	O(1)

The time complexity is O(1) because we only need to compute a small number of valid combinations based on the values of n and limit. Space complexity is also O(1) as we use a constant amount of space for calculations.

What Interviewers Usually Probe

Strong understanding of combinatorics and enumeration techniques.
Ability to optimize a brute-force solution for large input sizes.
Familiarity with problem-solving patterns involving distribution under constraints.

Common Pitfalls or Variants

Common pitfalls

Forgetting to account for the limit on the number of candies each child can receive.
Incorrectly assuming the problem can be solved through brute-force enumeration without optimization.
Misunderstanding the problem's constraints, leading to an inefficient or incorrect solution.

Follow-up variants

Distribute candies among a different number of children.
Distribute candies with varying limits per child.
Find the number of ways to distribute candies when each child must receive at least one candy.

FAQ

What is the most efficient way to solve the Distribute Candies Among Children II problem?

The most efficient way is to apply combinatorics to enumerate valid candy distributions while keeping track of the limit for each child, optimizing the process to avoid brute force.

How do I ensure that no child gets more than the limit number of candies?

The solution involves calculating the number of valid combinations where each child receives no more than the specified limit, using combinatorics and constraints to prune invalid combinations.

Can this problem be solved using brute force?

While brute force is possible, it is inefficient for large inputs. Optimizing the solution using combinatorics and dynamic programming is the recommended approach.

What mathematical concept should I focus on for this problem?

The problem primarily involves combinatorics and enumeration, where you need to count the valid distributions of candies under constraints.

How can I handle large inputs efficiently in this problem?

By using combinatorial counting methods and dynamic programming techniques, you can compute the result efficiently without the need for brute force enumeration.

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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