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.Text;
using System.Linq;

class Program
{
    public void Proc()
    {
        Reader.IsDebug = false;
        int[] inpt = Reader.ReadLine().Split(' ').Select(a => int.Parse(a)).ToArray();
        this.PointCount = inpt[0];
        int pathCount = inpt[1];

        for (int i = 0; i < pathCount; i++)
        {
            inpt = Reader.ReadLine().Split(' ').Select(a => int.Parse(a)).ToArray();

            if(!this.Path.ContainsKey(inpt[0])) {
                this.Path.Add(inpt[0], new Dictionary<int, bool>());
            }
            this.Path[inpt[0]].Add(inpt[1], true);
            if (!this.Path.ContainsKey(inpt[1]))
            {
                this.Path.Add(inpt[1], new Dictionary<int, bool>());
            }
            this.Path[inpt[1]].Add(inpt[0], true);
        }
        int ans = this.GetAns(1, 1);
        Console.WriteLine(ans);
    }

    private int GetAns(int pos, int flags)
    {
        if ((1 << PointCount) - 1 == flags)
        {
            return 1;
        }
        int ans = 0;
        if (this.Path.ContainsKey(pos))
        {
            foreach (int next in this.Path[pos].Keys)
            {
                int key = 1 << (next-1);
                if ((flags & key) > 0)
                {
                    continue;
                }
                ans += this.GetAns(next, flags + key);
            }
        }
        return ans;
    }

    private int PointCount = 0;

    private Dictionary<int, Dictionary<int, bool>> Path = new Dictionary<int, Dictionary<int, bool>>();


    public class Reader
    {
        public static bool IsDebug = true;
        private static String PlainInput = @"



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



";
        private static System.IO.StringReader Sr = null;
        public static string ReadLine()
        {
            if (IsDebug)
            {
                if (Sr == null)
                {
                    Sr = new System.IO.StringReader(PlainInput.Trim());
                }
                return Sr.ReadLine();
            }
            else
            {
                return Console.ReadLine();
            }
        }
    }
    static void Main()
    {
        Program prg = new Program();
        prg.Proc();
    }
}