CF1735D · Meta-set

• 难度：1700
• 标签：brute force、combinatorics、data structures、hashing、math
• 链接：https://codeforces.com/problemset/problem/1735/D
• 时间限制：4 seconds　内存限制：256 megabytes
• 出现位置：Day21-字符串综合-hash-二分-构造/选做

英文原题面

Statement

You like the card board game "Set". Each card contains $k$ features, each of which is equal to a value from the set $\{0, 1, 2\}$. The deck contains all possible variants of cards, that is, there are $3^k$ different cards in total. A feature for three cards is called good if it is the same for these cards or pairwise distinct. Three cards are called a set if all $k$ features are good for them. For example, the cards $(0, 0, 0)$, $(0, 2, 1)$, and $(0, 1, 2)$ form a set, but the cards $(0, 2, 2)$, $(2, 1, 2)$, and $(1, 2, 0)$ do not, as, for example, the last feature is not good. A group of five cards is called a meta-set, if there is strictly more than one set among them. How many meta-sets there are among given $n$ distinct cards?

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.

样例

样例 1

输入：

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

输出：

1

样例 2

输入：

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

输出：

3

样例 3

输入：

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

输出：

54

样例 4

输入：

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

输出：

0

样例解释（英文原文）

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.