The Chinese Remainder Theorem · Superchallenge problem

Find the remainder when $\dbinom{100}{50}$ is divided by $105$. (You may use Lucas's theorem: for a prime $p$, $\binom{m}{k} \equiv \prod_i \binom{m_i}{k_i} \pmod p$, where $m_i$, $k_i$ are the base-$p$ digits of $m$ and $k$.)