Peter Winkler's factorial problem

 A cute problem from Peter Winkler's collection of math puzzles.

$$S = \{n! | 1 \leq n \le 100, n \in N\}$$

Can we remove a single element from $S$ such that the product of the elements of the resulting set is a perfect square?

Scroll down for a solution.

Product of elements of $S$ is

$$P = 1!  2! \dots 99!  100!$$

Pair up terms $(2n-1)! (2n)! = ((2n-1)!)^2 2n$

Thus

$$P = (1!  3!  5! \dots 99!)^2 (2 . 4 . 6 \dots 100)$$

$$=  (1!  3!  5! \dots 99!)^2 . 2^{50} . 50!$$

Thus removing $50!$ from $S$ will give us the desired result.