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: Wrong Answer)

# 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, $array_b = $m.times.map{ gets.split.map(&:to_i) }.transpose
$array_a.freeze
$array_b.freeze

# count_1_a = $array_a.count(1)
# count_1_b = $array_b.count(1)

indexes_1_a = []
indexes_1_b = []

$array_a.each_with_index do |a, i|
  indexes_1_a << i if a == 1
end
$array_b.each_with_index do |b, i|
  indexes_1_b << i if b == 1
end

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

def move(to, reached_node, used_link)
  if reached_node.count(true) == $n
    $ans += 1
    return true
  end
  if used_link.length == $m
    return false
  end

  indexes_to_a = []
  indexes_to_b = []

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

  indexes_to_a.each do |i|
    to = $array_b[i]
    next if reached_node[to]
    reached_node[to] = true
    used_link << i
    move(to, reached_node, used_link)
  end
  indexes_to_b.each do |i|
    to = $array_a[i]
    next if reached_node[to]
    reached_node[to] = true
    used_link << i
    move(to, reached_node, used_link)
  end
end

indexes_1_a.each do |i|
  from = $array_a[i]
  to = $array_b[i]
  reached_node[from] = true
  reached_node[to] = true
  used_link << i
  move(to, reached_node, used_link)
end
indexes_1_b.each do |i|
  from = $array_b[i]
  to = $array_a[i]
  reached_node[from] = true
  reached_node[to] = true
  used_link << i
  move(to, reached_node, used_link)
end

puts $ans