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 PHP7 (7.0.15) (judge verdict: Wrong Answer)

<?php
$input = explode(' ', trim(fgets(STDIN)));
$n = (int)$input[0];
$m = (int)$input[1];

$vs = [];
for ($i=1; $i <= $n; $i++) { 
	$vs[$i]['color'] = 'w';
	$vs[$i]['path'] = [];
}

for ($j=1; $j <= $m ; $j++) { 
	$input = explode(' ', trim(fgets(STDIN)));
	$fr = (int)$input[0];
	$to = (int)$input[1];
	$vs[$fr]['path'][] = $to;
	$vs[$to]['path'][] = $fr;
}
// var_dump($vs);

$v_count = 1;
$vs[1]['color'] = 'b';
$count = 0;
dfs($vs[1]);

function dfs($v){
	global $vs;
	global $n;
	global $v_count;
	global $count;
	if ($n === $v_count) {
		++$count;
		return;
	}
	foreach ($v['path'] as $p) {
		if ($vs[$p]['color'] === 'w') {
			$vs[$p]['color'] = 'b';
			++$v_count;
			dfs($vs[$p]);
			$vs[$p]['color'] === 'w';
			--$v_count;
		}
	}
	return;
}

echo $count;