Factorial Product Is Square
Find the largest integer \(n \leq 200\) such that for some positive integer \(k< n\) the product \(n!k!\) is a perfect square.
Details and assumptions
The number \( n!\), read as n factorial, is equal to the product of all positive integers less than or equal to \(n\). For example, \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
If you think that no such \(n\) exists, type your answer as 0.