Is Distribute Candies Among Children III easy or hard?

The problem is rated Hard because recognizing the combinatorial structure is not obvious. Once you identify the stars and bars formulation and apply inclusion-exclusion to enforce the limit constraint, the implementation becomes short and efficient.

Distribute Candies Among Children III Python/Java solution

The implementation computes combinations such as C(n+2,2) and adjusts counts using inclusion-exclusion for each child exceeding the limit. The logic is identical across Python, Java, C++, Go, and TypeScript and runs in constant time with simple arithmetic operations.

How to solve Distribute Candies Among Children III in O(1)?

Treat the distribution as the equation a + b + c = n. Without constraints the number of solutions is C(n+2,2). Then apply inclusion-exclusion to remove cases where a, b, or c exceed the limit by transforming variables such as a' = a − (limit+1). Evaluating these combinations directly gives a constant-time solution.

What is the best approach for Distribute Candies Among Children III?

The optimal approach uses combinatorial mathematics with the principle of inclusion-exclusion. First count all solutions to a + b + c = n using the stars and bars formula C(n+2,2). Then subtract distributions where a child exceeds the limit and correct overlaps using inclusion-exclusion. This computes the result in O(1) time and O(1) space.

Is Distribute Candies Among Children III asked at Google/Amazon/Meta?

Combinatorics and counting problems using stars and bars or inclusion-exclusion frequently appear in interviews at companies like Google, Amazon, and Meta. While the exact problem may vary, the mathematical technique used here is a common interview pattern.

What data structure is used in Distribute Candies Among Children III?

No special data structure is required. The problem relies purely on mathematical counting using combinations and the inclusion-exclusion principle. The implementation typically uses a helper function to compute nCk values.

What is the time complexity of Distribute Candies Among Children III?

The optimal combinatorial solution runs in O(1) time because it only evaluates a small number of combination formulas. No iteration over candies or limits is required. Space complexity is also O(1) since only a few integer calculations are performed.

Description Hints 5 Solution Video 35:23

2927. Distribute Candies Among Children III

Hard57.3% acceptance2 companiesPremium elsewhere, free here

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 <= 10⁸
1 <= limit <= 10⁸

Asked at 2 companies

Rubrik Amazon

Topics

Combinatorics Math

Hints open one at a time

Hint 1

Try to solve the problem using combinatorics.

Hint 2

If the limit didn’t exist, the problem would be distributing n candies between 3 children.

Hint 3

The answer to the above problem would be C(n + 2, 2).

Hint 4

Now try to combine this with the Inclusion-exclusion principle.

Hint 5

Apart from the above solution, there are other mathematical solutions that you can think of.

35:23

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3
3

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