LeetCode #2928 — EASY

Distribute Candies Among Children I

Build confidence with an intuition-first walkthrough focused on math fundamentals.

The Problem

Problem Statement

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.

Example 1:

Input: n = 5, limit = 2
Output: 3
Explanation: 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
Explanation: 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

Step 01

Brute Force Baseline

Problem summary: 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.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Math

Example 1

5
2

Example 2

3
3

Core Insight

What unlocks the optimal approach

Use three nested for loops to check all the triplets.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2928: Distribute Candies Among Children I
class Solution {
    public int distributeCandies(int n, int limit) {
        if (n > 3 * limit) {
            return 0;
        }
        long ans = comb2(n + 2);
        if (n > limit) {
            ans -= 3 * comb2(n - limit + 1);
        }
        if (n - 2 >= 2 * limit) {
            ans += 3 * comb2(n - 2 * limit);
        }
        return (int) ans;
    }

    private long comb2(int n) {
        return 1L * n * (n - 1) / 2;
    }
}

// Accepted solution for LeetCode #2928: Distribute Candies Among Children I
func distributeCandies(n int, limit int) int {
	comb2 := func(n int) int {
		return n * (n - 1) / 2
	}
	if n > 3*limit {
		return 0
	}
	ans := comb2(n + 2)
	if n > limit {
		ans -= 3 * comb2(n-limit+1)
	}
	if n-2 >= 2*limit {
		ans += 3 * comb2(n-2*limit)
	}
	return ans
}

# Accepted solution for LeetCode #2928: Distribute Candies Among Children I
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

// Accepted solution for LeetCode #2928: Distribute Candies Among Children I
fn distribute_candies(n: i32, limit: i32) -> i32 {
    fn f(n: i32, people: i32, limit: i32) -> i32 {
        if people >= 3 {
            if n == 0 {
                return 1;
            }

            return 0;
        }

        let mut ret = 0;
        let limit = std::cmp::min(n, limit);
        for i in 0..=limit {
            ret += f(n - i, people + 1, limit);
        }

        ret
    }

    f(n, 0, limit)
}

fn main() {
    let ret = distribute_candies(3, 3);
    println!("ret={ret}");
}

#[test]
fn test() {
    assert_eq!(distribute_candies(5, 2), 3);
    assert_eq!(distribute_candies(3, 3), 10);
}

// Accepted solution for LeetCode #2928: Distribute Candies Among Children I
function distributeCandies(n: number, limit: number): number {
    const comb2 = (n: number) => (n * (n - 1)) / 2;
    if (n > 3 * limit) {
        return 0;
    }
    let ans = comb2(n + 2);
    if (n > limit) {
        ans -= 3 * comb2(n - limit + 1);
    }
    if (n - 2 >= 2 * limit) {
        ans += 3 * comb2(n - 2 * limit);
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(1)

Space

O(1)

Approach Breakdown

ITERATIVE

O(n) time

O(1) space

Simulate the process step by step — multiply n times, check each number up to n, or iterate through all possibilities. Each step is O(1), but doing it n times gives O(n). No extra space needed since we just track running state.

MATH INSIGHT

O(log n) time

O(1) space

Math problems often have a closed-form or O(log n) solution hidden behind an O(n) simulation. Modular arithmetic, fast exponentiation (repeated squaring), GCD (Euclidean algorithm), and number theory properties can dramatically reduce complexity.

Shortcut: Look for mathematical properties that eliminate iteration. Repeated squaring → O(log n). Modular arithmetic avoids overflow.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.