For what values of n and r, a binomial coefficient C(n,r) is a perfect square?

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R. C. Mittal

For C(n, n/2) to be perfect square, 0

David G Glynn

Bertrand's postulate (proved by Chebyshev in 1852) states that there is prime between x and 2x-2 for x>3. This means that C(n,n/2) = n(n-1)...(n/2+1)/n/2.(n/2-1)...2.1 has a single prime factor bigger than n/2 and less than n-2, and so, after checking a couple of small cases it can't be a perfect square.

https://en.wikipedia.org/wiki/Bertrand%27s_postulate