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 C (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <memory.h>

int N, M;
bool *gone, *used;
int **side;

int culc_pass(int top, int count);

int main(void)
{
  int i, answer;


  /*入力*/
  scanf("%d %d", &N, &M);

  /*辺の格納用の動的配列の作成*/
  side = (int**)malloc(sizeof(int*) * M);
  if (side == NULL) printf("MEMORY ERROR OF side");
  for (i=0;i<M;i++) {
    side[i] = (int*)malloc(sizeof(int) * 2);
    if (side[i] == NULL) printf("MEMORY ERROR OF side %d", i);
  }

  /*辺の入力*/
  for (i=0;i<M;i++) {
    scanf("%d %d", &side[i][0], &side[i][1]);
  }

  /*行った頂点を記憶する配列と、使った辺を記憶する配列を作成*/
  gone = (bool*)malloc(sizeof(bool) * N);
  if (gone == NULL) printf("MEMORY ERROR OF gone");
  used = (bool*)malloc(sizeof(bool) * M);
  if (used == NULL) printf("MEMORY ERROR OF used");

  /*初期値をfalseに設定*/
  for (i=0;i<N;i++) {
    gone[i] = false;
  }
  for (i=0;i<M;i++) {
    used[i] = false;
  }

  /*探索*/
  answer = culc_pass(1, 1);

  /*出力*/
  printf("%d\n", answer);

  /*メモリ解放*/
  free(gone);
  free(used);

  for (i=0;i<M;i++) {
    free(side[i]);
  }
  free(side);

  return 0;
}


int culc_pass(int top, int count)
{
  int i, sum = 0;
  bool temp_used[28], temp_gone[8];
  /*全部いけたらおｋ*/
  if (count == N) return 1;

  /*つぎいけるとこを探索*/
  for (i=0;i<M;i++) {
    /*頂点が今いるとこじゃなかったらだめ*/
    if (side[i][0] != top) continue;
    /*行ったことあったらだめ*/
    if (gone[side[i][1]]) continue;
    /*一回通った道だったらだめ*/
    if (used[i]) continue;
    /*いけるとこの結果を足し合わせる*/
    used[i] = true;
    gone[side[i][1]] = true;
    memcpy(temp_used, used, sizeof(used));
    memcpy(temp_gone, gone, sizeof(gone));
    sum = sum + culc_pass(side[i][1], count + 1);
    memcpy(used, temp_used, sizeof(used));
    memcpy(gone, temp_gone, sizeof(gone));
  }
  return sum;
}