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Find the Number of Possible Ways for an Event

Number: 3604

Difficulty: Hard

Paid? No

Companies: Amazon

Problem Description

There are n performers and x stages. Each performer is assigned to one of the x stages. Performers assigned to the same stage form a band, and empty stages are allowed. After all performances, every non‐empty band receives a score in the range [1, y]. Two events are considered different if at least one performer is assigned to a different stage or if any band (non‐empty stage) is awarded a different score. The task is to calculate the total number of possible events modulo 10^9+7.

Key Insights

  • The assignment of performers to stages is independent of the awarding of scores.
  • For each assignment, only the occupied (non‐empty) stages receive a score.
  • If exactly k stages are non‐empty then there are (binom(x, k) * ways to assign n performers to get exactly k non‐empty stages) ways to assign performers, and each occupied stage can receive y scores.
  • Use dynamic programming to compute the number of performer assignments that yield exactly j non‐empty stages, then multiply by y^j for the score choices.
  • Recurrence: dp[i+1][j] = j * dp[i][j] (assign performer to an already occupied stage) + (x - j) * dp[i][j] (assign performer to a new stage).

Space and Time Complexity

Time Complexity: O(nx), due to the nested loops over the number of performers and stages. Space Complexity: O(nx) if using a 2D dp table; this can be optimized to O(x) using a single array.

Solution

We solve the problem using dynamic programming. Let dp[i][j] represent the number of ways to assign i performers such that exactly j stages are occupied. When adding a new performer, we have two options:

  1. Put the performer into one of the already occupied j stages (which gives j choices).
  2. Assign the performer to one of the remaining (x-j) stages (which will result in a new non‐empty stage).

After processing all performers, for each scenario with j occupied stages, we have y^j ways to award scores to these bands. Sum these results for j from 1 to x to get the final answer (modulo 10^9+7).

Code Solutions

# Python solution

MOD = 10**9 + 7

def findWays(n, x, y):
    # dp[j] will be the number of ways for current performer count with j non-empty stages.
    dp = [0] * (x + 1)
    dp[0] = 1  # With 0 performers, 0 stages are used.
    
    for i in range(n):
        # We prepare next state dp_next for the next performer.
        dp_next = [0] * (x + 1)
        for j in range(x + 1):
            if dp[j]:
                # Option 1: assign to an already used stage (if there are j ways)
                dp_next[j] = (dp_next[j] + dp[j] * j) % MOD
                # Option 2: assign to a new stage (if available)
                if j < x:
                    dp_next[j + 1] = (dp_next[j + 1] + dp[j] * (x - j)) % MOD
        dp = dp_next

    # Modular exponentiation for y^j
    answer = 0
    power_y = [1] * (x + 1)
    for j in range(1, x + 1):
        power_y[j] = (power_y[j - 1] * y) % MOD

    # Sum up dp[j] * y^(j) for all j >= 1 (since j = 0 means no band is awarded)
    for j in range(1, x + 1):
        answer = (answer + dp[j] * power_y[j]) % MOD

    return answer

# Example Test Cases
print(findWays(1, 2, 3))  # Expected output: 6
print(findWays(5, 2, 1))  # Expected output: 32
print(findWays(3, 3, 4))  # Expected output: 684
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