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.Text;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using static System.Console;
using static System.Math;

namespace NotFounds
{
    public class Program
    {
        public static void Main(string[] args)
        {
            new Program().Solve();
        }

        public Graph g;
        public void Solve()
        {
            var cin = new MyInputStream();
            var n = cin.ri();
            var m = cin.ri();

            visited = new bool[n];
            g = new Graph(n);
            for (int i = 0; i < m; ++i)
            {
                var a = cin.ri() - 1;
                var b = cin.ri() - 1;
                g.AddUnDirectedEdge(a, b);
            }

            WriteLine(dfs(0));
        }

        private bool[] visited;
        public int dfs(int node)
        {
            if (visited[node]) return 0;
            else visited[node] = true;
            if (check())
            {
                visited[node] = false;
                return 1;
            }
            var sum = 0;
            foreach (var n in g.g[node])
                sum += dfs(n.To);
            visited[node] = false;
            return sum;
        }

        public bool check()
        {
            return visited.All(x => x);
        }
    }

    public class Graph
    {
        public struct Edge
        {
            public int To;
            public long Cost;
            public Edge(int t, long c)
            {
                this.To = t;
                this.Cost = c;
            }
        }

        public struct Node : IComparable<Node>
        {
            public int Pos;
            public long Cost;
            public Node(int p, long c)
            {
                this.Pos = p;
                this.Cost = c;
            }
            public int CompareTo(Node obj)
            {
                return (int)(this.Cost - obj.Cost);
            }
        }

        public List<List<Edge>> g;
        public int N { get; set; }
        public Graph(int size)
        {
            N = size;
            g = new List<List<Edge>>(size);
            for (int i = 0; i < size; i++) g.Add(new List<Edge>());
        }

        public void AddDirectedEdge(int from, int to, long cost = 0)
        {
            g[from].Add(new Edge(to, cost));
        }

        public void AddUnDirectedEdge(int from, int to, int cost = 0)
        {
            AddDirectedEdge(from, to, cost);
            AddDirectedEdge(to, from, cost);
        }
    }

    public class MyInputStream
    {
        private char separator = ' ';
        private Queue<string> inputStream;
        public MyInputStream(char separator = ' ')
        {
            this.separator = separator;
            inputStream = new Queue<string>();
        }

        public string Read()
        {
            if (inputStream.Count != 0) return inputStream.Dequeue();
            string[] tmp = Console.ReadLine().Trim().Split(separator);
            for (int i = 0; i < tmp.Length; i++)
            {
                inputStream.Enqueue(tmp[i]);
            }
            return inputStream.Dequeue();
        }
        public string ReadLine() => Console.ReadLine();
        public int ReadInt() => int.Parse(Read());
        public long ReadLong() => long.Parse(Read());
        public double ReadDouble() => double.Parse(Read());
        public string r() => Read();
        public int ri() => ReadInt();
        public long rl() => ReadLong();
        public double rd() => ReadDouble();
    }
}