Is there any perfect squares that are also binomial coefficients?
It is well known that the equation $$ \binom{n}{k}=m^{\ell} $$ has no integer solutions with $ℓ ≥ 2$ and $4 ≤ k ≤ n − 4$. For $k=3$ and $\ell=2$ we only have the solution $\binom{50}{3}=140^2$, and for $k=\ell=2$ there are more. This is due to Erdős. For details see "Binomial coefficients are (almost) never powers" in the book of proofs by Aigner and Ziegler.