Committee With a Quota

Question

A desk has 8 traders and 6 quants. How many 5-person committees contain at least two quants?

Accepted Answer

Complement count. All committees minus the bad ones (0 or 1 quant): binom145 - binom85 - binom61binom84 = 2002 - 56 - 420 = 1526. Approach 2, direct sum Add up exactly k quants for k = 2,3,4,5: binom62binom83 + binom63binom82 + binom64binom81 + binom65binom80 = 840 + 560 + 120 + 6 = 1526. checkmark The trap interviewers are fishing for: "choose 2 quants to satisfy the quota, then any 3 of the remaining 12" gives binom62binom123 = 3300, wrong because a committee with 3 quants gets counted binom32 = 3 times, once for each pair that could have been the "quota pair." Sequential choice + an "at least" constraint = systematic overcounting. Complement (or exact-case sums) is the clean route, and saying why the shortcut fails is worth as much as the number.

Committee With a Quota

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