Score : $100$ points

Problem Statement

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:

$N$

$a_{1, 1}$ $\ldots$ $a_{1, N}$

$:$

$a_{N, 1}$ $\ldots$ $a_{N, N}$

Output

Print the number of ways in which Taro can make $N$ pairs, modulo $10^9 + 7$.

3
0 1 1
1 0 1
1 1 1

3

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

• $(1, 2), (2, 1), (3, 3)$
• $(1, 2), (2, 3), (3, 1)$
• $(1, 3), (2, 1), (3, 2)$

4
0 1 0 0
0 0 0 1
1 0 0 0
0 0 1 0

1

There is one way to make pairs, as follows:

• $(1, 2), (2, 4), (3, 1), (4, 3)$

1
0

0

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

102515160

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