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: Wrong Answer)

using System;
class Program
{

static string[] input = Console.ReadLine().Split(' ');//Splitで区切り文字を指定して複数個受け取る。
static int n = int.Parse(input[0]);//頂点数
static int m = int.Parse(input[1]);//辺数
static int[,] path = new int[n,n];//隣接行列
static bool[] complete = new bool[n];//通ったらtrue。初期値false
static int completeNum = 1;//すでに通った頂点数
static int answer = 0;

	static void Main()
	{
    for(int i = 0; i < m; i++)//隣接行列の完成
    {
    int[] nums = Array.ConvertAll(Console.ReadLine().Split(' '),int.Parse);
    if(nums[0] <= nums[1]) path[nums[0]-1, nums[1]-1]++;
    else path[nums[1]-1, nums[0]-1]++;
    }

    complete[0] = true;
    CheckPath(0);//頂点0と繋がる頂点へ進む
    
	Console.WriteLine(answer);
	}

  static void CheckPath(int start)//引数startと繋がる頂点へ進む
  {
    for(int i = 1; i < n; i++)
    {
      if(path[start, i] == 1 && !(complete[i]))
      {
        complete[i] = true;
        completeNum++;
        Console.WriteLine(start + " " + i);
        if(completeNum == n) answer++;
        else CheckPath(i);
        complete[i] = false;
        completeNum--;
      }
    }
  
  }
  
}