Is it realy probality?

Write the positive integers in a $x^2y$ formula, where $x$ is a positive integer, and $y$ is a square free positive integer. (Square free means that except 1 it doesn't have a divisor, which is a square number.) Looking at the first $25$ positive integer, the possible values of $y$ are:

$1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23$

This is $16$ numbers. So if we choose $17$ numbers, then there will be two numbers, which $y$ 's are equal, so the product of them will be a square number.

So the answer is $\boxed{1}$ .