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.split(' ').map(&:to_i)
$edges = []
$m.times do 
  $edges << gets.split(' ').map(&:to_i).sort
end	
$nodes = $edges.flatten.uniq.sort

def find_edges(node, edges)
  edges.select{|e| e[0] == node} + edges.select{|e| e[1] == node}.map{|e| [e[1], e[0]]}
end

def traverse(start_node, visited_nodes, used_edges, previous_edge)

  used_edges << previous_edge.sort unless previous_edge.nil?
  visited_nodes << start_node
    puts "#{start_node} #{visited_nodes} #{used_edges}"
  if visited_nodes.length == $n
    puts "GOAL"
    return
  end
  if used_edges == $m
    puts "FAILED"
    return
  end
  find_edges(start_node, $edges - used_edges).each do |next_edge|
    puts "next: #{next_edge} used: #{used_edges}"
    traverse(next_edge[1], visited_nodes.dup, used_edges.dup, next_edge)
  end
end

p $nodes
p $edges
traverse(1, [], [], nil)