100542: [AtCoder]ABC054 C - One-stroke Path

Problem Statement

You are given an undirected unweighted graph with $N$ vertices and $M$ edges that contains neither self-loops nor double edges.
Here, a self-loop is an edge where $a_i = b_i (1≤i≤M)$, and double edges are two edges where $(a_i,b_i)=(a_j,b_j)$ or $(a_i,b_i)=(b_j,a_j) (1≤i<j≤M)$.
How many different paths start from vertex $1$ and visit all the vertices exactly once?
Here, the endpoints of a path are considered visited.

For example, let us assume that the following undirected graph shown in Figure 1 is given.

Figure 1: an example of an undirected graph

The following path shown in Figure 2 satisfies the condition.

Figure 2: an example of a path that satisfies the condition

However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.

Figure 3: an example of a path that does not satisfy the condition

Neither the following path shown in Figure 4, because it does not start from vertex $1$.

Figure 4: another example of a path that does not satisfy the condition

Constraints

• $2≦N≦8$
• $0≦M≦N(N-1)/2$
• $1≦a_i<b_i≦N$
• The given graph contains neither self-loops nor double edges.

Input

The input is given from Standard Input in the following format:

$N$ $M$  
$a_1$ $b_1$  
$a_2$ $b_2$
$:$  
$a_M$ $b_M$  

Output

Print the number of the different paths that start from vertex $1$ and visit all the vertices exactly once.

Sample Input 1

3 3
1 2
1 3
2 3

Sample Output 1

2

The given graph is shown in the following figure:

The following two paths satisfy the condition:

Sample Input 2

7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7

Sample Output 2

1

This test case is the same as the one described in the problem statement.