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 OCaml (4.02.3) (judge verdict: Accepted)

let n, m = Scanf.scanf "%d %d" (fun x y -> x, y)
let graph = Array.init n (fun _ -> [])
let _ = Array.init m (fun _ -> Scanf.scanf " %d %d" (fun a b ->
  graph.(a-1) <- b-1 :: graph.(a-1);
  graph.(b-1) <- a-1 :: graph.(b-1)))
let visited = Array.make n false
let rec dfs i d =
  if d >= n then 1 else
    graph.(i) |> List.map (fun j ->
      if visited.(j) then 0 else (
        visited.(j) <- true;
        let v = dfs j (d+1) in
        visited.(j) <- false;
        v
      )
    ) |> List.fold_left (+) 0

let () =
  visited.(0) <- true;
  dfs 0 1 |> Printf.printf "%d\n"