Find the Number of Possible Ways for an Event

Problem Description

You are given three integers n, x, and y. An event is being held for n performers. When a performer arrives, they are assigned to one of the x stages. All performers assigned to the same stage will perform together as a band, though some stages might remain empty. After all performances are completed, the jury will award each band a score in the range [1, y]. Return the total number of possible ways the event can take place. Since the answer may be very large, return it modulo 10^9 + 7.

Key Insights

• Each performer can be assigned to one of the x stages, leading to x^n possible assignments.
• Each band created (which corresponds to a unique stage) can receive a score from 1 to y, resulting in y^k possible score combinations, where k is the number of non-empty stages.
• The total number of configurations must consider all possible non-empty stage combinations, which can be calculated using the principle of inclusion-exclusion.
• The final answer is calculated as the product of the total assignments and the score configurations, adjusted for the empty stages.

Space and Time Complexity

Time Complexity: O(n * x)
Space Complexity: O(1)

Solution

To solve the problem, we first need to calculate the number of ways to assign n performers to x stages, which is x^n. Next, we need to determine the number of ways to score the bands formed by the assigned performers. This requires calculating how many non-empty subsets of stages can be formed. We can use the principle of inclusion-exclusion to count all possible combinations of scores based on the number of non-empty stages.

Steps:

1. Calculate total_assignments = x^n % (10^9 + 7).
2. For each possible number of non-empty stages k (from 1 to x), calculate the number of ways to choose k stages from x and the number of ways to assign scores from 1 to y to those k stages.
3. Sum the results for k from 1 to x.

Code Solutions