Problem:
Score : 300 points

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.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
//using System.Numerics;
class Program
{     
    static int AllPathThroughCount(List<int>[] node, int index, bool[] through)
    {
        if(through[index])
            return 0;
        through[index] = true;
        if(through.All(x => x))
        {
            through[index] = false;
            return 1;
        }
        int total = node[index]
            .Where(x => !through[x])
            .Select(x => AllPathThroughCount(node, x, through))
            .Sum();
        through[index] = false;
        return total;
    }
    public void Slove()
    {
        var inputs = Console.ReadLine().Split();
        int n = int.Parse(inputs[0]);
        int m = int.Parse(inputs[1]);
        var node = 
            Enumerable.Range(0, n)
            .Select(x => new List<int>())
            .ToArray();
        foreach(int i in Enumerable.Range(0, m))
        {
            inputs = Console.ReadLine().Split();
            int a = int.Parse(inputs[0]) - 1;
            int b = int.Parse(inputs[1]) - 1;
            node[a].Add(b);
            node[b].Add(a);
        }
        int result = AllPathThroughCount(node, 0, new bool[n]);
        System.Console.WriteLine(result);        
    }
    
    static void Main(string[] args)
    {
        new Program().Slove();
    }
}