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;

public static partial class Program
{
    public static long n;
    public static bool[,] graph = new bool[8, 8];
    public static void Main()
    {
        var nm = CRL().ArrayLongs();
        n = nm[0];
        for (int i = 0; i < nm[1]; i++)
        {
            var ab = CRL().ArrayLongs().Select(x => x - 1).ToArray();
            graph[ab[0], ab[1]] = true;
            graph[ab[1], ab[0]] = true;
        }

        var passed = new bool[n, n];
        var visited = new bool[n];
        visited[0] = true;
        CWL(DFS(0, passed, visited));
    }

    public static int DFS(int vert, bool[,] passed, bool[] visited)
    {
        if (visited.All(x => x))
        {
            return 1;
        }

        var ret = 0;
        for (int next = 0; next < n; next++)
        {
            //道が無かったら
            if (graph[vert, next] == false)
            {
                continue;
            }

            //既に通った道だったら
            if (passed[vert, next])
            {
                continue;
            }

            //既に訪れていたら
            if (visited[next])
            {
                continue;
            }

            visited[next] = true;      //頂点
            passed[vert, next] = true; //道(進むとき)
            passed[next, vert] = true; //道(戻るとき)
            ret += DFS(next, passed, visited);
            visited[next] = false;
            passed[vert, next] = false;
            passed[next, vert] = false;
        }

        return ret;
    }
}

public static partial class Program
{
    //テンプレここから
    private static long ToInt(this string str) => long.Parse(str);
    private static long[] ToInt(this IEnumerable<string> strs) => strs.Select(x => x.ToInt()).ToArray();
    public static long[] ArrayLongs(this string str, char opr = ' ') => str.Split(opr).ToInt();
    public static string CRL() => Console.ReadLine();
    public static void CWL(object obj) => Console.WriteLine(obj);
    public static long Diff(long a, long b) => Math.Abs(Math.Max(a, b) - Math.Min(a, b));
    public static long Sign(this long i) => Math.Sign(i);
    public static bool IsEven(this long i) => i % 2 == 0;
    public static bool IsEven(this int i) => i % 2 == 0;
    public static long[] dx = new long[] { 0, -1, 0, 1 };
    public static long[] dy = new long[] { -1, 0, 1, 0 };
    public static long mod = 1000000007;
    public static string alphabet = "abcdefghijklmnopqrstuvwxyz";
    public static long ToLong(this char c) => c.ToString().ToInt();
    //テンプレここまで
}