Erdős and Graham write that 'a proof that this cannot happen infinitely often for $\binom{n}{2}$ seems hopeless; probably this can never happen for $\binom{n}{k}$ if $3\leq k\leq n-3$.'

Weisenberg has provided four easy examples that show Erdős and Graham were too optimistic here: \[\binom{7}{3}=5\cdot 7,\] \[\binom{10}{4}= 2\cdot 3\cdot 5\cdot 7,\] \[\binom{14}{4} = 7\cdot 11\cdot 13,\] and \[\binom{15}{6}=5\cdot 7\cdot 11\cdot 13.\]