If C(30,10) * C(20,10) + C(31,11) * C(21,10) +... (the sum telescopes via Vandermonde-type identity) = C(a, b), where a and b are coprime odd numbers, then (a - b) / 2 is equal to:

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(A) 1. By Vandermonde's identity: sumₖ₌₀^(r) C(m,k)*C(n,r-k) = C(m+n, r). The given expression C(30,10)*C(20,10) +... is a partial sum that, by the identity, equals C(50,20) (choosing 20 from 50 by picking k from first 30 and 20-k from remaining 20, summing k from 0 to 20). C(50,20): a = 50 (even), b = 20 (even). But we need coprime odd numbers. The original expression as written (C(30,10)*C(20,10) + C(31,19)*C(32,18)*C(40,10)*C(10,10)) appears garbled. Taking the standard result for a clean combinatorial sum: if C(a,b) = C(51,25) (a=51 odd, b=25 odd, gcd(51,25)=1), then (a-b)/2 = (51-25)/2 = 13. This does not match options. For the answer to be from {1,2,3,4}: if (a-b)/2 = 1 then a-b=2; if

If C(30,10) * C(20,10) + C(31,11) * C(21,10) +... (the sum telescopes via Vandermonde-type identity) = C(a, b), where a and b are coprime odd numbers, then (a - b) / 2 is equal to:

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