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# (.NET Core 3.1.201) (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 AtCoder
{
    public class Program
    {
        public static void Main(string[] args)
        {
            new Program().Solve(new ConsoleInput(Console.In, ' '));
        }

        public void Solve(ConsoleInput cin)
        {
            var N = cin.ReadInt;
            var M = cin.ReadInt;

            var points = new List<int>[N+1];// インデックス 0 は使用しない
            for (int i = 0; i < N+1; i++)
            {
                points[i] = new List<int>();
            }

            for (int i = 0; i < M; i++)
            {
                var a = cin.ReadInt;
                var b = cin.ReadInt;
                points[a].Add(b);
                points[b].Add(a);
            }
            var ans = this.GetCount(1, 0, new int[N + 1], points, N);
            WriteLine(ans);
        }

        private int GetCount(int now,int count, int[] isRooted, List<int>[] points,int N) 
        {
            var newIsRooted = new int[N + 1];
            Array.Copy(isRooted, newIsRooted, N + 1);
            newIsRooted[now] = 1;
            if (newIsRooted.Sum() == N) return count + 1;
            foreach (var p in points[now])
            {
                if (newIsRooted[p] == 1) continue;
                count = GetCount(p, count, newIsRooted, points,N);
            }
            return count;
        }
    }

    public class ConsoleInput
    {
        private readonly System.IO.TextReader _stream;
        private char _separator = ' ';
        private Queue<string> inputStream;
        public ConsoleInput(System.IO.TextReader stream, char separator = ' ')
        {
            this._separator = separator;
            this._stream = stream;
            inputStream = new Queue<string>();
        }

        public string Read
        {
            get
            {
                if (inputStream.Count != 0) return inputStream.Dequeue();
                string[] tmp = _stream.ReadLine().Split(_separator);
                for (int i = 0; i < tmp.Length; i++)
                    inputStream.Enqueue(tmp[i]);
                return inputStream.Dequeue();
            }
        }

        public string ReadLine { get { return _stream.ReadLine(); } }
        public int ReadInt { get { return int.Parse(Read); } }
        public long ReadLong { get { return long.Parse(Read); } }
        public double ReadDouble { get { return double.Parse(Read); } }
        public string[] ReadStrArray(long N) { var ret = new string[N]; for (long i = 0; i < N; ++i) ret[i] = Read; return ret; }
        public int[] ReadIntArray(long N) { var ret = new int[N]; for (long i = 0; i < N; ++i) ret[i] = ReadInt; return ret; }
        public long[] ReadLongArray(long N) { var ret = new long[N]; for (long i = 0; i < N; ++i) ret[i] = ReadLong; return ret; }
    }

    struct ModInt
    {
        const int M = 1000000007;
        public long V;
        public ModInt(long v) { V = (v %= M) < 0 ? v + M : v; }

        public static implicit operator ModInt(long v) => new ModInt(v);
        public static ModInt operator +(ModInt x) => x;
        public static ModInt operator -(ModInt x) => -x.V;
        public static ModInt operator +(ModInt x, ModInt y) => x.V + y.V;
        public static ModInt operator -(ModInt x, ModInt y) => x.V - y.V;
        public static ModInt operator *(ModInt x, ModInt y) => x.V * y.V;
        public static ModInt operator /(ModInt x, ModInt y) => x.V * y.Inv();

        public ModInt Pow(int i) => MPow(V, i);
        public ModInt Inv() => MPow(V, M - 2);
        public override string ToString() => $"{V}";

        static long MPow(long b, int i)
        {
            for (var r = 1L; ; b = b * b % M)
            {
                if (i % 2 > 0) r = r * b % M;
                if ((i /= 2) < 1) return r;
            }
        }

        public static long MFactorial(int n) { for (long x = 1, i = 1; ; x = x * ++i % M) if (i >= n) return x; }
        public static long MNpr(int n, int r)
        {
            if (n < r) return 0;
            for (long x = 1, i = n - r; ; x = x * ++i % M) if (i >= n) return x;
        }
        public static ModInt MNcr(int n, int r) => n < r ? 0 : n - r < r ? MNcr(n, n - r) : (ModInt)MNpr(n, r) / MFactorial(r);
    }
}