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++14 (GCC 5.4.1) (judge verdict: Accepted)

// In the name of Allah the Most Merciful.

#include<bits/stdc++.h>
using namespace std;

typedef long long ll;
ll dp[(1<<8)+10][10];
int Set(int n , int pos)
{
    return n|(1<<pos);
}
bool Check(int n , int pos)
{
    return n&(1<<pos);
}
vector<int>edges[10];
int n , m;

ll f(int mask , int u)
{
    if(mask==(1<<n)-1)return 1;
    if(dp[mask][u]!=-1)return dp[mask][u];
    ll value = 0;
    for(auto v:edges[u]){
        if(Check(mask , v)==false){
            value+=f(Set(mask , v) , v);
        }
    }
    return dp[mask][u] = value;
}

int main(void)
{
    ios::sync_with_stdio(0);
    cin.tie(0);cout.tie(0);

    cin >> n >> m;
    for(int i=0;i<m;i++){
        int in1 , in2;
        cin >> in1 >> in2;
        in1--;
        in2--;
        edges[in1].push_back(in2);
        edges[in2].push_back(in1);
    }
    memset(dp , -1 , sizeof(dp));
    ll ans = 0;
    int mask = Set(0 , 0);
    for(int i=0;i<edges[0].size();i++){
        ans+=f(Set(mask , edges[0][i]) , edges[0][i]);
    }
    cout << ans << endl;
    return 0;
}