Find the Number of Possible Ways for an Event

Find the Number of Possible Ways for an Event

What is this problem about?

The Find the Number of Possible Ways for an Event interview question is a combinatorial task. You are given $n$ items (or people) and $m$ categories (or rooms). You need to find the number of ways to assign each item to a category such that a specific constraint is met (e.g., at least one item per category, or exactly $k$ categories are used). This often involves Stirling Numbers of the Second Kind.

Why is this asked in interviews?

Amazon uses this "Hard" problem to test a candidate's mastery of Combinatorics and Dynamic Programming. It evaluation whether you can break a large counting problem into smaller, recursive sub-problems and whether you understand how to use inclusion-exclusion or Stirling numbers to handle "unlabeled" vs "labeled" sets.

Algorithmic pattern used

This problem is solved using Dynamic Programming (Stirling Numbers) or Inclusion-Exclusion.

1. Let dp[i][j] be the number of ways to distribute $i$ distinct items into $j$ distinct non-empty categories.
2. Transition: For the next item, you can either:
   • Put it in one of the $j$ categories already in use: $j imes dp[i-1][j]$.
   • Put it in a brand new category: $dp[i-1][j-1]$.
3. The result for exactly $k$ categories is $\binom{m}{k} imes dp[n][k]$.
4. Apply modulo $10^9 + 7$ at each step.

Example explanation

$n=3$ items, $m=2$ categories.

1. Ways to use exactly 1 category: $\binom{2}{1} imes 1^3 = 2$. (All in A, all in B).
2. Ways to use exactly 2 categories: $\binom{2}{2} imes ( ext{Stirling } S(3,2) imes 2!) = 1 imes (3 imes 2) = 6$. Total ways = $2 + 6 = 8$. (Which matches $2^3$, every item picking one of 2 categories).

Common mistakes candidates make

• Ignoring non-empty rule: Failing to account for the fact that some categories might be empty if the problem requires they be filled.
• Permutation error: Forgetting to multiply by $k!$ if the categories are distinct.
• Integer Overflow: Not using long integers or applying the modulo correctly during the power calculations.

Interview preparation tip

Practice the Combinatorics interview pattern. Know the formulas for combinations, permutations, and Stirling numbers. Being able to explain the "item-by-item" transition in a DP table is a high-level skill that demonstrates deep logical thinking.