Score : $300$ points

问题描述

设有 $M$ 个集合，分别记作 $S_1, S_2, \dots, S_M$，包含从 $1$ 到 $N$ 的整数。
其中，集合 $S_i$ 包含 $C_i$ 个整数 $a_{i, 1}, a_{i, 2}, \dots, a_{i, C_i}$。

从这 $M$ 个集合中选择一个或多个集合的方式有 $(2^M-1)$ 种。
其中满足以下条件的组合方式有多少种？

• 对于所有满足 $1 \leq x \leq N$ 的整数 $x$，至少有一个被选中的集合包含 $x$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $M$ sets, called $S_1, S_2, \dots, S_M$, consisting of integers between $1$ and $N$.
$S_i$ consists of $C_i$ integers $a_{i, 1}, a_{i, 2}, \dots, a_{i, C_i}$.

There are $(2^M-1)$ ways to choose one or more sets from the $M$ sets.
How many of them satisfy the following condition?

• For all integers $x$ such that $1 \leq x \leq N$, there is at least one chosen set containing $x$.

Constraints

• $1 \leq N \leq 10$
• $1 \leq M \leq 10$
• $1 \leq C_i \leq N$
• $1 \leq a_{i,1} \lt a_{i,2} \lt \dots \lt a_{i,C_i} \leq N$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$C_1$

$a_{1,1}$ $a_{1,2}$ $\dots$ $a_{1,C_1}$

$C_2$

$a_{2,1}$ $a_{2,2}$ $\dots$ $a_{2,C_2}$

$\vdots$

$C_M$

$a_{M,1}$ $a_{M,2}$ $\dots$ $a_{M,C_M}$

Output

Print the number of ways to choose sets that satisfy the conditions in the Problem Statement.

Sample Input 1

3 3
2
1 2
2
1 3
1
2

Sample Output 1

3

The sets given in the input are $S_1 = \lbrace 1, 2 \rbrace, S_2 = \lbrace 1, 3 \rbrace, S_3 = \lbrace 2 \rbrace$.
The following three ways satisfy the conditions in the Problem Statement:

• choosing $S_1, S_2$;
• choosing $S_1, S_2, S_3$;
• choosing $S_2, S_3$.

Sample Input 2

4 2
2
1 2
2
1 3

Sample Output 2

0

There may be no way to choose sets that satisfies the conditions in the Problem Statement.

Sample Input 3

6 6
3
2 3 6
3
2 4 6
2
3 6
3
1 5 6
3
1 3 6
2
1 4

Sample Output 3

18

update @ 2024/3/10 12:04:22