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 (2.7.1) (judge verdict: Accepted)

# 3 3   n: node, m: link
# 1 2   a_1   b_1
# 1 3
# 2 3   a_m   b_m

$n, $m = gets.split.map(&:to_i)
array_a_temp, array_b_temp = $m.times.map{ gets.split.map(&:to_i) }.transpose
$array_a = array_a_temp + array_b_temp
$array_b = array_b_temp + array_a_temp

reached_node = Array.new($n, false)
used_link = []
$ans = 0

# $binding = 0

def move(to, reached_node, used_link)
  # $binding += 1
  # puts "binding: #{$binding}"
  # p reached_node
  # p used_link
  if reached_node.count(true) == $n
    $ans += 1
    return true
  end
  if used_link.length == $m
    return false
  end

  indexes_to_a = []
  $array_a.each_with_index do |a, i|
    indexes_to_a << i if a == to && !(used_link.include?(i))
  end

  indexes_to_a.each do |i|
    reached_node_temp = []
    reached_node.each do |j|
      reached_node_temp << j
    end
    used_link_temp = []
    used_link.each do |j|
      used_link_temp << j
    end
    to = $array_b[i]
    next if reached_node[to-1]
    reached_node_temp[to-1] = true
    used_link_temp << i
    move(to, reached_node_temp, used_link_temp)
  end
end

indexes_1_a = []
$array_a.each_with_index do |a, i|
  indexes_1_a << i if a == 1
end

indexes_1_a.each do |i|
  reached_node = Array.new($n, false)
  from = $array_a[i]
  to = $array_b[i]
  reached_node[from-1] = true
  reached_node[to-1] = true
  used_link = [i]
  move(to, reached_node, used_link)
end

puts $ans