Meta-set

Description:

Input:

The first line of the input contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 10^3$$$, $$$1 \le k \le 20$$$) — the number of cards on a table and the number of card features. The description of the cards follows in the next $$$n$$$ lines.

Each line describing a card contains $$$k$$$ integers $$$c_{i, 1}, c_{i, 2}, \ldots, c_{i, k}$$$ ($$$0 \le c_{i, j} \le 2$$$) — card features. It is guaranteed that all cards are distinct.

Output:

Output one integer — the number of meta-sets.

Sample Input:

8 4
0 0 0 0
0 0 0 1
0 0 0 2
0 0 1 0
0 0 2 0
0 1 0 0
1 0 0 0
2 2 0 0

Sample Output:

1

Sample Input:

7 4
0 0 0 0
0 0 0 1
0 0 0 2
0 0 1 0
0 0 2 0
0 1 0 0
0 2 0 0

Sample Output:

3

Sample Input:

9 2
0 0
0 1
0 2
1 0
1 1
1 2
2 0
2 1
2 2

Sample Output:

54

Sample Input:

20 4
0 2 0 0
0 2 2 2
0 2 2 1
0 2 0 1
1 2 2 0
1 2 1 0
1 2 2 1
1 2 0 1
1 1 2 2
1 1 0 2
1 1 2 1
1 1 1 1
2 1 2 0
2 1 1 2
2 1 2 1
2 1 1 1
0 1 1 2
0 0 1 0
2 2 0 0
2 0 0 2

Sample Output:

0

Note:

Let's draw the cards indicating the first four features. The first feature will indicate the number of objects on a card: $$$1$$$, $$$2$$$, $$$3$$$. The second one is the color: red, green, purple. The third is the shape: oval, diamond, squiggle. The fourth is filling: open, striped, solid.

You can see the first three tests below. For the first two tests, the meta-sets are highlighted.

In the first test, the only meta-set is the five cards $$$(0000,\ 0001,\ 0002,\ 0010,\ 0020)$$$. The sets in it are the triples $$$(0000,\ 0001,\ 0002)$$$ and $$$(0000,\ 0010,\ 0020)$$$. Also, a set is the triple $$$(0100,\ 1000,\ 2200)$$$ which does not belong to any meta-set.

In the second test, the following groups of five cards are meta-sets: $$$(0000,\ 0001,\ 0002,\ 0010,\ 0020)$$$, $$$(0000,\ 0001,\ 0002,\ 0100,\ 0200)$$$, $$$(0000,\ 0010,\ 0020,\ 0100,\ 0200)$$$.

In there third test, there are $$$54$$$ meta-sets.