Distribute Candies Among Children III

Distribute Candies Among Children III

What is this problem about?

The Distribute Candies Among Children III coding problem is the "Hard" version of the series. While the logic remains identical to version II (distributing $n$ candies to 3 children with a limit), the constraints are pushed to the extreme ($n$ and limit up to $10^9$). The requirement is a strictly $O(1)$ solution using pure mathematical formulas.

Why is this asked in interviews?

Rubrik and Amazon ask this to test your absolute mastery of Combinatorics. It evaluations whether you can correctly implement the Principle of Inclusion-Exclusion without any loops or recursion. It’s a test of precision—handling large-scale arithmetic where every term must be calculated using a constant-time formula.

Algorithmic pattern used

The pattern is Combinatorics (Stars and Bars) with Inclusion-Exclusion.

1. Define a helper function ways(count) which returns $\binom{count+2}{2}$ if count >= 0, and 0 otherwise.
2. Calculate the four components of PIE:
   • A = ways(n)
   • B = 3 * ways(n - (limit + 1))
   • C = 3 * ways(n - 2 * (limit + 1))
   • D = ways(n - 3 * (limit + 1))
3. Result = $A - B + C - D$.

Example explanation

$n=10, limit=3$.

1. ways(10) = 12 * 11 / 2 = 66.
2. ways(10 - 4) = 6. ways(6) = 8 * 7 / 2 = 28. $3 \times 28 = 84$.
3. ways(10 - 8) = 2. ways(2) = 4 * 3 / 2 = 6. $3 \times 6 = 18$.
4. ways(10 - 12) = -2. Returns 0. Result: $66 - 84 + 18 - 0 = 0$. (If $n=10$ and each can only take 3, the max total is 9, so 0 ways is correct).

Common mistakes candidates make

• Handling Large N: Forgetting that $n^2$ can be $10^{18}$, which requires 64-bit integers.
• Boundary conditions: Not properly returning 0 for the ways function when the required candies exceed $n$.
• Conceptual confusion: Mixing up "identical items" (candies) with "distinct items," which would require different formulas.

Interview preparation tip

Practice the Principle of Inclusion-Exclusion for 3 sets. It is a common pattern for "at most" constraints. The general formula for $k$ bins is $\sum_{i=0}^{k} (-1)^i \binom{k}{i} \binom{n - i(limit+1) + k - 1}{k - 1}$.