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 5.4.1) (judge verdict: Accepted)

#include <stdio.h>

int N, M, adj[10][10] = {}, bit[10] = {0, 0, 1, 2, 4, 8, 16, 32, 64}, dp[10][128];

int recursion(int n, int rest)
{
	if (dp[n][rest] >= 0) return dp[n][rest];
	else if (rest == bit[n]) {
		dp[n][rest] = 1;
		return dp[n][rest];
	} else dp[n][rest] = 0;
	
	int i;
	for (i = 2; i <= N; i++) {
		if (adj[n][i] == 1 && (rest | bit[i]) == rest) dp[n][rest] += recursion(i, rest ^ bit[n]);
	}
	return dp[n][rest];
}

int main()
{
	int i, u, w;
	scanf("%d %d", &N, &M);
	for (i = 1; i <= M; i++) {
		scanf("%d %d", &u, &w);
		adj[u][w] = 1;
		adj[w][u] = 1;
	}
	
	int j, ans = 0, all = 1 << (N - 1);
	for (i = 2; i <= N; i++) for (j = 1; j <= 127; j++) dp[i][j] = -1;
	for (i = 2; i <= N; i++) if (adj[1][i] == 1) ans += recursion(i, all - 1);
	printf("%d\n", ans);
	fflush(stdout);
	return 0;
}