Distribute Candies Among Children I

Distribute Candies Among Children I

What is this problem about?

The Distribute Candies Among Children I coding problem asks you to find the total number of ways to distribute $n$ identical candies among 3 children such that no child receives more than limit candies. Every candy must be given to a child. For example, if $n=3$ and limit=3, there are 10 ways (including combinations like (3,0,0), (1,1,1), etc.).

Why is this asked in interviews?

Companies like Rubrik and Amazon use this problem to test a candidate's basic combinatorial skills and their ability to implement nested loops. Since the constraints for "Version I" are usually small ($n \le 50$), a brute-force approach is expected. It evaluations your ability to translate a word problem into a controlled search space.

Algorithmic pattern used

The primary pattern here is Enumeration (Brute Force). You can use two nested loops to represent the number of candies given to the first and second children.

1. child1 goes from 0 to limit.
2. child2 goes from 0 to limit.
3. The number of candies for the third child is child3 = n - child1 - child2.
4. If child3 is within the range [0, limit], you have found a valid distribution.

Example explanation

$n=2, limit=1$

1. Child 1 = 0, Child 2 = 0 $\to$ Child 3 = 2 (Invalid, $2 > 1$).
2. Child 1 = 0, Child 2 = 1 $\to$ Child 3 = 1 (Valid: 0, 1, 1).
3. Child 1 = 1, Child 2 = 0 $\to$ Child 3 = 1 (Valid: 1, 0, 1).
4. Child 1 = 1, Child 2 = 1 $\to$ Child 3 = 0 (Valid: 1, 1, 0). Total ways: 3.

Common mistakes candidates make

• Loop Boundaries: Forgetting that a child can receive 0 candies.
• Condition checks: Not checking if child3 is non-negative before checking the limit.
• Inefficient counting: Trying to use recursion when two simple loops are sufficient for these constraints.

Interview preparation tip

For problems involving distributing $N$ items into $K$ bins with small $N$, nested loops are your best friend. However, always mention that for larger $N$, you would look into Stars and Bars or the Principle of Inclusion-Exclusion to optimize the solution.