Social Distance 2

AtCoder

IDabc225_h

Time2000ms

Memory256MB

Difficulty—

There are $N$ chairs arranged in a row, called Chair $1$, Chair $2$, $\ldots$, Chair $N$. A chair seats only one person. $M$ people will sit on $M$ of these chairs. Here, let us define the score as follows: > $\displaystyle \prod_{i=1}^{M-1} (B_{i+1} - B_i)$, where $B=(B_1,B_2,\ldots,B_M)$ is the sorted list of the indices of the chairs the people sit on. Person $i$ $(1 \leq i \leq K)$ is already sitting on Chair $A_i$. There are ${} _ {N-K} \mathrm{P} _ {M-K}$ ways for the other $M-K$ people to take seats. Find the sum of the scores for all of these ways. Since this sum may be enormous, compute it modulo $998244353$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $2 \leq M \leq N$ * $0 \leq K \leq M$ * $1 \leq A_1 \lt A_2 \lt \ldots \lt A_K \leq N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ $A_1$ $A_2$ $\ldots$ $A_K$ [samples]

Samples

Input #1

5 3 2
1 3

Output #1

7

If Person $3$ sits on Chair $2$, the score will be $(2-1) \times (3-2)=1 \times 1 = 1$.  
If Person $3$ sits on Chair $4$, the score will be $(3-1) \times (4-3)=2 \times 1 = 2$.  
If Person $3$ sits on Chair $5$, the score will be $(3-1) \times (5-3)=2 \times 2 = 4$.  
The answer is $1+2+4=7$.

Input #2

6 6 1
4

Output #2

120

The score for every way of sitting will be $1$.  
There are ${} _ {5} \mathrm{P} _ {5} = 120$ ways of sitting, so the answer is $120$.

Input #3

99 10 3
10 50 90

Output #3

761621047

API Response (JSON)

{
  "problem": {
    "name": "Social Distance 2",
    "description": {
      "content": "There are $N$ chairs arranged in a row, called Chair $1$, Chair $2$, $\\ldots$, Chair $N$.   A chair seats only one person. $M$ people will sit on $M$ of these chairs. Here, let us define the score as ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc225_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ chairs arranged in a row, called Chair $1$, Chair $2$, $\\ldots$, Chair $N$.  \nA chair seats only one person.\n$M$ people will sit on $M$ of these chairs. Here, let us define the score as ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments