Lottery

AtCoder

IDabc243_f

Time2000ms

Memory256MB

Difficulty—

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 $998244353$. ## 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: $N$ $M$ $K$ $W_1$ $\vdots$ $W_N$ [samples] ## 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 $xz\equiv y \pmod{998244353}$.

Samples

Input #1

2 1 2
2
1

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 $221832079$.

Input #2

Output #2

0

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

Input #3

Output #3

335346748

Input #4

Output #4

755239064

API Response (JSON)

{
  "problem": {
    "name": "Lottery",
    "description": {
      "content": "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 d",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc243_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi is participating in a lottery.\nEach 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 d...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments