Min Product Sum

AtCoder

IDagc039_f

Time6000ms

Memory256MB

Difficulty—

For each of the $K^{NM}$ ways to write an integer between $1$ and $K$ (inclusive) in every square in a square grid with $N$ rows and $M$ columns, find the value defined below, then compute the sum of all those $K^{NM}$ values, modulo $D$. * For each of the $NM$ squares, find the minimum among the $N+M-1$ integers written in the square's row or the square's column. The value defined for the grid is the product of all these $NM$ values. ## Constraints * $1 \leq N,M,K \leq 100$ * $10^8 \leq D \leq 10^9$ * $N,M,K,$ and $D$ are integers. * $D$ is prime. ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ $D$ [samples]

Samples

Input #1

2 2 2 998244353

Output #1

35

We have $1$ way to write integers such that the product of the $NM$ values is $16$, $4$ ways such that the product is $2$, and $11$ ways such that the product is $1$.

Input #2

2 3 4 998244353

Output #2

Input #3

31 41 59 998244353

Output #3

827794103

API Response (JSON)

{
  "problem": {
    "name": "Min Product Sum",
    "description": {
      "content": "For each of the $K^{NM}$ ways to write an integer between $1$ and $K$ (inclusive) in every square in a square grid with $N$ rows and $M$ columns, find the value defined below, then compute the sum of ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc039_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For each of the $K^{NM}$ ways to write an integer between $1$ and $K$ (inclusive) in every square in a square grid with $N$ rows and $M$ columns, find the value defined below, then compute the sum of ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments