Exhaustive enumeration: for all $4\le n\le 5\times10^6$ and all $2\le k\le n/2$ ($6{,}249{,}995{,}000{,}001$ pairs), the only binomial coefficients equal to a product of consecutive primes are the nine known solutions $(n,k)=(4,2),(6,2),(7,3),(10,4),(14,4),(15,2),(15,6),(21,2),(715,2)$. Additionally no solution with $k\le 64$ exists up to $n=10^7$, extending the published $k=2$ search frontier (A280992: $5\times10^6$) to $10^7$.