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 Python (3.4.3) (judge verdict: Runtime Error)

import copy


class Line:
    def __init__(self, a, b):
        self.a = a
        self.b = b


def generate_patterns(remains, current_pattern, patterns):
    if len(remains) == 0:
        patterns.append(current_pattern)
        return
    for remain in remains:
        _current_pattern = copy.copy(current_pattern)
        _current_pattern.append(remain)
        _remains = copy.copy(remains)
        _remains.remove(remain)
        generate_patterns(_remains, _current_pattern, patterns)
    return patterns


N, M = map(int, input().split())
lines = []
for _ in range(N):
    a, b = map(int, input().split())
    lines.append(Line(a, b))

m = {}
for i in range(1, N+1):
    m[i] = {}
    for j in range(1, N+1):
        m[i][j] = False

for line in lines:
    m[line.a][line.b] = True
    m[line.b][line.a] = True

remains = []
for i in range(2, N+1):
    remains.append(i)
patterns = generate_patterns(remains, [1], [])

cnt = 0
for pattern in patterns:
    countable = True
    for i in range(len(pattern)-1):
        current = pattern[i]
        next = pattern[i+1]
        if not(m[current][next]):
            countable = False
            break
    if countable:
        cnt += 1

print(cnt)