#3317

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Find the Number of Possible Ways for an Event

Given n performers, x stages, and y scores, calculate the number of possible ways to assign performers and score bands.

Problem Statement

You are tasked with organizing an event with n performers. There are x stages where each performer must be assigned to a stage. Performers assigned to the same stage will perform as a band, and some stages may remain empty. After all performances, the jury will assign a score from 1 to y to each band formed by performers on the same stage.

Your goal is to find how many different ways this event can occur, considering both the assignments of performers to stages and the scores assigned to the bands formed by the performers on each stage.

Examples

Example 1

Input: n = 1, x = 2, y = 3

Output: 6

Example 2

Input: n = 5, x = 2, y = 1

Output: 32

Example 3

Input: n = 3, x = 3, y = 4

Output: 684

Example details omitted.

Constraints

1 <= n, x, y <= 1000

Solution Approach

Dynamic Programming Transition

Use a dynamic programming approach where the state represents the number of ways to assign performers to a set number of stages. Transition the state by iterating over all possible ways to assign each performer to one of the x stages and applying the possible score range y to each band.

Combinatorics for Assignments

For each performer, calculate the number of ways they can be assigned to one of the stages. Use combinatorics to determine the number of ways to distribute performers across stages and calculate the score for each stage's band.

Efficient State Calculation

To optimize, use dynamic programming to store previously computed states and avoid redundant calculations. This reduces the complexity significantly and allows handling the problem constraints effectively.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the dynamic programming approach chosen. Typically, the complexity will involve iterating over the n performers and x stages, with the number of transitions growing in proportion to the input values.

What Interviewers Usually Probe

Candidate demonstrates knowledge of state transition dynamic programming.
Candidate efficiently handles combinatorics for stage assignments.
Candidate can optimize using dynamic programming to handle large inputs.

Common Pitfalls or Variants

Common pitfalls

Not properly handling the stage assignment combinatorics.
Overcomplicating the dynamic programming state transitions.
Failing to optimize the solution for larger inputs, especially when n, x, and y grow.

Follow-up variants

Use dynamic programming with memoization for larger values of n and x.
Consider using matrix exponentiation to speed up transitions.
Modify the problem to allow for stage weights or score ranges outside of [1, y].

FAQ

What is the primary algorithmic pattern in this problem?

The primary pattern is state transition dynamic programming, which tracks the number of ways to assign performers to stages and assign scores to bands.

How can I handle large inputs efficiently?

To handle large inputs, use dynamic programming with memoization to store intermediate results and avoid recalculating states.

What combinatorics are involved in this problem?

The combinatorics involve determining how to distribute n performers across x stages and then assigning scores from the range [1, y] to each band formed.

Why should I fix the number of stages in the approach?

Fixing the number of stages simplifies the dynamic programming transitions and ensures that the solution remains efficient, even with the larger input constraints.

What is the key difficulty of this problem?

The key difficulty lies in efficiently calculating the number of ways to assign performers and their scores, while managing the complexity of dynamic programming transitions.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j]$ to represent the number of ways to arrange the first $i$ performers into $j$ programs. Initially, $f[0][0] = 1$, and the rest $f[i][j] = 0$.

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class Solution:
    def numberOfWays(self, n: int, x: int, y: int) -> int:
        mod = 10**9 + 7
        f = [[0] * (x + 1) for _ in range(n + 1)]
        f[0][0] = 1
        for i in range(1, n + 1):
            for j in range(1, x + 1):
                f[i][j] = (f[i - 1][j] * j + f[i - 1][j - 1] * (x - (j - 1))) % mod
        ans, p = 0, 1
        for j in range(1, x + 1):
            p = p * y % mod
            ans = (ans + f[n][j] * p) % mod
        return ans

Continue Practicing

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