E - K-colinear Line

Score : $500$ points

Problem Statement

You are given $N$ points in the coordinate plane. For each $1\leq i\leq N$, the $i$-th point is at the coordinates $(X_i, Y_i)$.

Find the number of lines in the plane that pass $K$ or more of the $N$ points.
If there are infinitely many such lines, print Infinity.

Constraints

• $1 \leq K \leq N \leq 300$
• $\lvert X_i \rvert, \lvert Y_i \rvert \leq 10^9$
• $X_i\neq X_j$ or $Y_i\neq Y_j$, if $i\neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_N$ $Y_N$

Output

Print the number of lines in the plane that pass $K$ or more of the $N$ points, or Infinity if there are infinitely many such lines.

5 2
0 0
1 0
0 1
-1 0
0 -1

6

The six lines $x=0$, $y=0$, $y=x\pm 1$, and $y=-x\pm 1$ satisfy the requirement.
For example, $x=0$ passes the first, third, and fifth points.

Thus, $6$ should be printed.

1 1
0 0

Infinity

Infinitely many lines pass the origin.

Thus, Infinity should be printed.