Coincidence

AtCoder

ID1202Contest_d

Time2000ms

Memory256MB

Difficulty—

You have a pair of integer sequences $A=(A_1, A_2, \dots, A_N), B=(B_1, B_2, \dots, B_N)$. Initially $A_i=B_i=0$ holds for all $i = 1, 2, \dots, N$. You will perform the following operation $M$ times. * **Operation**: Choose two integers $i, j\ (1 \le i, j \le N)$ and increment $A_i$ and $B_j$ by $1$. Here, out of the $M$ operations, the number of operations in which $i=j$ holds must be **exactly** $X$. Find the number, modulo $998244353$, of pairs of integer sequences that $A, B$ can be after the $M$ operations. ## Constraints * $2 \leq N \leq 3000$ * $0 \leq X \leq M \le 3000$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N \ M \ X$ [samples]

Samples

Input #1

3 1 1

Output #1

3

The following $3$ pairs are possible.

*   $A=(1,0,0), \ B=(1,0,0)$
*   $A=(0,1,0), \ B=(0,1,0)$
*   $A=(0,0,1), \ B=(0,0,1)$

Input #2

3 1 0

Output #2

6

The following $6$ pairs are possible.

*   $A=(1,0,0), \ B=(0,1,0)$
*   $A=(1,0,0), \ B=(0,0,1)$
*   $A=(0,1,0), \ B=(1,0,0)$
*   $A=(0,1,0), \ B=(0,0,1)$
*   $A=(0,0,1), \ B=(1,0,0)$
*   $A=(0,0,1), \ B=(0,1,0)$

Input #3

4 4 2

Output #3

643

The followings are examples of possible pairs.

*   $A=(1,1,1,1), \ B=(1,1,1,1)$
*   $A=(1,0,0,3), \ B=(0,1,0,3)$

Input #4

314 1592 653

Output #4

755768689

API Response (JSON)

{
  "problem": {
    "name": "Coincidence",
    "description": {
      "content": "You have a pair of integer sequences $A=(A_1, A_2, \\dots, A_N), B=(B_1, B_2, \\dots, B_N)$. Initially $A_i=B_i=0$ holds for all $i = 1, 2, \\dots, N$. You will perform the following operation $M$ times.",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "1202Contest_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You have a pair of integer sequences $A=(A_1, A_2, \\dots, A_N), B=(B_1, B_2, \\dots, B_N)$. Initially $A_i=B_i=0$ holds for all $i = 1, 2, \\dots, N$.\nYou will perform the following operation $M$ times....",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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