Score : $500$ points

问题陈述

高桥正在参加一个彩票抽奖活动。

每次抽奖时，他都会获得 $N$ 个奖品中的一个。第 $i$ 个奖品被抽中的概率为 $\frac{W_i}{\sum_{j=1}^{N}W_j}$。每次抽奖的结果都是相互独立的。

请问，在进行 $K$ 次抽奖后，他恰好获得 $M$ 个不同奖品的概率是多少？请计算结果并以 $998244353$ 取模。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Takahashi is participating in a lottery.

Each time he takes a draw, he gets one of the $N$ prizes available. Prize $i$ is awarded with probability $\frac{W_i}{\sum_{j=1}^{N}W_j}$. The results of the draws are independent of each other.

What is the probability that he gets exactly $M$ different prizes from $K$ draws? Find it modulo $998244353$.

Note

To print a rational number, start by representing it as a fraction $\frac{y}{x}$. Here, $x$ and $y$ should be integers, and $x$ should not be divisible by $998244353$ (under the Constraints of this problem, such a representation is always possible). Then, print the only integer $z$ between $0$ and $998244352$ (inclusive) such that $xz\equiv y \pmod{998244353}$.

Constraints

• $1 \leq K \leq 50$
• $1 \leq M \leq N \leq 50$
• $0 < W_i$
• $0 < W_1 + \ldots + W_N < 998244353$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

$W_1$

$\vdots$

$W_N$

Output

Print the answer.

Sample Input 1

2 1 2
2
1

Sample Output 1

221832079

Each draw awards Prize $1$ with probability $\frac{2}{3}$ and Prize $2$ with probability $\frac{1}{3}$.

He gets Prize $1$ at both of the two draws with probability $\frac{4}{9}$, and Prize $2$ at both draws with probability $\frac{1}{9}$, so the sought probability is $\frac{5}{9}$.

The modulo $998244353$ representation of this value, according to Note, is $221832079$.

Sample Input 2

3 3 2
1
1
1

Sample Output 2

0

It is impossible to get three different prizes from two draws, so the sought probability is $0$.

Sample Input 3

3 3 10
499122176
499122175
1

Sample Output 3

335346748

Sample Input 4

10 8 15
1
1
1
1
1
1
1
1
1
1

Sample Output 4

755239064

update @ 2024/3/10 10:27:43