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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define FOR(i, a, b) for(ll i = (a); i < (b); ++i)
#define REP(i, n) for(ll i = 0; i < (n); ++i)
#define ARRAY_LENGTH(array) sizeof(array)/sizeof(*array)
template<class T> inline bool chmin(T& a, T b) { if (a > b) { a = b; return true; } return false; }
template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return true; } return false; }

int main() {
    ll N, M;
    cin >> N >> M;

    vector<int> sides[N + 1];
    REP(i, M) {
        int a, b;
        cin >> a >> b;
        sides[a].push_back(b);
        sides[b].push_back(a);
    }

    vector<int> one_case;
    for(int i = 2; i <= N; ++i) {
        one_case.emplace_back(i);
    }

    int wayCnt = 0;
    do {
        bool canGoThrough = true;
        int cr = 1, nx = one_case[0], nx_num = 0;

        do { 
            bool canMove = false;

            for(int j = 0; j < (int)sides[cr].size(); ++j) {
                if(sides[cr][j] == one_case[nx_num]) {
                    canMove = true;
                }
            }

            if(!canMove) {
                canGoThrough = false;
            }

            if(nx_num != N - 2) {
                cr = nx;
                ++nx_num;
                nx = one_case[nx_num];
            } 
        } while(nx_num != N - 2 && canGoThrough);

        if(canGoThrough) {
            ++wayCnt;
        }
    } while(next_permutation(one_case.begin(), one_case.end()));

    cout << wayCnt << endl;

    return 0;
}