Matching

AtCoder

IDdp_o

Time2000ms

Memory256MB

Difficulty—

There are $N$ men and $N$ women, both numbered $1, 2, \ldots, N$. For each $i, j$ ($1 \leq i, j \leq N$), the compatibility of Man $i$ and Woman $j$ is given as an integer $a_{i, j}$. If $a_{i, j} = 1$, Man $i$ and Woman $j$ are compatible; if $a_{i, j} = 0$, they are not. Taro is trying to make $N$ pairs, each consisting of a man and a woman who are compatible. Here, each man and each woman must belong to exactly one pair. Find the number of ways in which Taro can make $N$ pairs, modulo $10^9 + 7$. ## Constraints * All values in input are integers. * $1 \leq N \leq 21$ * $a_{i, j}$ is $0$ or $1$. ## Input Input is given from Standard Input in the following format: $N$ $a_{1, 1}$ $\ldots$ $a_{1, N}$ $:$ $a_{N, 1}$ $\ldots$ $a_{N, N}$ [samples]

Samples

Input #1

Output #1

3

There are three ways to make pairs, as follows ($(i, j)$ denotes a pair of Man $i$ and Woman $j$):

*   $(1, 2), (2, 1), (3, 3)$
*   $(1, 2), (2, 3), (3, 1)$
*   $(1, 3), (2, 1), (3, 2)$

Input #2

Output #2

1

There is one way to make pairs, as follows:

*   $(1, 2), (2, 4), (3, 1), (4, 3)$

Input #3

1
0

Output #3

Input #4

21
0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 0 1
1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0
0 0 1 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 1 1
0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0
1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 0 1
0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0
0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1
0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1
0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1
0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0
0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1
0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1
1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1
0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1
1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1
0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1
0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0
1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 0
1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0

Output #4

102515160

Be sure to print the number modulo $10^9 + 7$.

API Response (JSON)

{
  "problem": {
    "name": "Matching",
    "description": {
      "content": "There are $N$ men and $N$ women, both numbered $1, 2, \\ldots, N$. For each $i, j$ ($1 \\leq i, j \\leq N$), the compatibility of Man $i$ and Woman $j$ is given as an integer $a_{i, j}$. If $a_{i, j} = 1",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "dp_o"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ men and $N$ women, both numbered $1, 2, \\ldots, N$.\nFor each $i, j$ ($1 \\leq i, j \\leq N$), the compatibility of Man $i$ and Woman $j$ is given as an integer $a_{i, j}$. If $a_{i, j} = 1...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments