Factorial Product Is Square

Number Theory Level 3

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.