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: Runtime Error)

#!/usr/bin/ruby
#encoding: utf-8

# DFS and the number of path of graph
# https://book.mynavi.jp/manatee/detail/id=65960
# ABC 054 C

# input 
$n,m = gets.split.map(&:to_i)
a = Array.new(m)
b = Array.new(m)
(0..m-1).each do |i|
  a[i],b[i] = gets.split.map(&:to_i)
  a[i] -= 1
  b[i] -= 1
end
$used = Array.new($n,0)
$connect = Array.new($n).map{Array.new($n,0)}

(0..$n-1).each do |i|
  # if connected 1, else 0
  $connect[a[i]][b[i]] = 1
  $connect[b[i]][a[i]] = 1
end

# dfs
def dfs(now,depth)
  # return 0 if used.
  return 0 if $used[now] == 1

  # this path is effective if the depth is equal to n.
  return 1 if depth == $n 

  $used[now] = 1
  ans = 0

  # check all connected vertices from "now".
  (0..$n-1).each do |i|
    if $connect[now][i] == 1 then
      ans += dfs(i,depth+1)
   end 
  end

  $used[now] = 0
  return ans
end  

# main
puts dfs(0,1)