Expand It Out?

By Kummer's Theorem , the Binomial coefficient $\binom{m+n}{m}$ is odd precisely when there are no "carries" when $m$ and $n$ are added in base $2$ . Since the binary expansion of $1000$ is $1111101000_2$ , the only way that $\binom{1000}{m}$ can be odd is if the binary expansion of $m$ is of the form $a_1a_2a_3a_4a_50a_6000_2 \hspace{2cm} a_1,a_2,a_3,a_4,a_5,a_6 \in \{0,1\}$ so that the binary expansion of $1000-m$ is $b_1b_2b_3b_4b_50b_6000_2 \hspace{2cm} b_j = 1-a_j \; (1 \le j \le 6)$ Thus there are $2^6 = 64$ numbers $m$ between $0$ and $1000$ for which $\binom{1000}{m}$ is odd, and hence $\boxed{937}$ even Binomial coefficients.

The coefficients are the 1001 numbers $\binom{1000}{k}$ with k = 0, 1, ... , 1000. By Lucas's theorem , the coefficient is odd if no binary digit of k is greater than the corresponding digit of 1000. We have 1000 = 1111101000 in binary. This means that k takes the form xxxxx0x000 in binary, where each x is a binary digit, either 1 or 0, but not all of them are the same. This is easily computed as $2^6 = 64$ odd coefficients, so there are 937 even ones.