Binomial Coefficient Even or Odd?

Question

How to check whether the value of binomial coefficient nCr is even or odd ?

Answer 1

Hint: You can use the Vandermonde convolution $\binom{n}{k}=\sum_{j=0}^k \binom{m}{j}\binom{n-m}{k-j}$ (for any $0\le m\le n$) for the case when $n$ is even (choose $m=n/2$) (you will notice that $\binom{n}{k}$ is even when $k$ is odd in this case). When $n$ is odd, you can use a similar strategy.

But, of course, as in the links provided in the comments, you may also use the more involved theorem due to Kummer.