The Majority

AtCoder

IDarc134_c

Time2000ms

Memory256MB

Difficulty—

There are $K$ boxes numbered $1$ to $K$. Initially, all boxes are empty. Snuke has some balls with integers from $1$ to $N$ written on them. Among them, $a_i$ balls have the number $i$. Balls with the same integer written on them cannot be distinguished. Snuke has decided to put all of his balls in the boxes. He wants the balls with the number $1$ to have a majority in every box. In other words, in every box, the number of balls with $1$ should be greater than that of the other balls. Find the number of such ways to put the balls in the boxes, modulo $998244353$. Two ways are distinguished when there is a pair of integers $(i,j)$ satisfying $1 \leq i \leq K, 1 \leq j \leq N$ such that the numbers of balls with $j$ in Box $i$ are different. ## Constraints * All values in input are integers. * $1 \leq N \leq 10^5$ * $1 \leq K \leq 200$ * $1 \leq a_i < 998244353$ ## Input Input is given from Standard Input in the following format: $N$ $K$ $a_1$ $\cdots$ $a_N$ [samples]

Samples

Input #1

2 2
3 1

Output #1

2

*   There are two ways to put the balls so that the balls with the number $1$ will be in the majority.
*   One way to do so is to put two of the balls with $1$ in Box $1$ and the other one in Box $2$, and put the one ball with $2$ in Box $1$.

Input #2

2 1
1 100

Output #2

0

*   There may be no way to put the balls in the boxes to meet the requirement.

Input #3

20 100
1073813 90585 41323 52293 62633 28788 1925 56222 54989 2772 36456 64841 26551 92115 63191 3603 82120 94450 71667 9325

Output #3

313918676

*   Be sure to find the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "The Majority",
    "description": {
      "content": "There are $K$ boxes numbered $1$ to $K$. Initially, all boxes are empty. Snuke has some balls with integers from $1$ to $N$ written on them. Among them, $a_i$ balls have the number $i$. Balls with the",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc134_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $K$ boxes numbered $1$ to $K$. Initially, all boxes are empty.\nSnuke has some balls with integers from $1$ to $N$ written on them. Among them, $a_i$ balls have the number $i$. Balls with the...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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