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 Ruby (2.3.3) (judge verdict: Wrong Answer)

n, m = gets.chomp.split(' ').map(&:to_i)
path = []

m.times do
  path << gets.chomp.split(' ').map(&:to_i)
end

points = path.flatten.uniq

if points.size < n - 1
  puts 0
  exit(0)
end

def dame(n)
  res = 0
  n.each do |l|
    res += 1 if l.first == 1
    return true if res == 2
  end
  false
end

def search(points, start)
  points.each do |p|
    return p if p[0] == start || p[1] == start
  end
  nil
end

res = 0

path.combination(n - 1) do |p|
  next if dame(p)
  p = p.sort

  next if p.first.first.to_i != 1

  t = p.shift[1]
  until p.empty?
    point = search(p, t)
    break unless point
    t = point[0] == t ? point[1] : point[0]
    p.delete(point)
  end
  res += 1 if p.empty?
end

puts res