I can't seem to find anything in the literature on how to estimate the g.c.d. of $\binom{a}{k}$ and $\binom{b}{k}$. In particular, I would like to know why $\gcd(\binom{a}{3}, \binom{b}{3})\leq b \binom{a-b}{3}$ for $a-b\geq 4$. I'm confident it's true (computer search).