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A music director must form a 5-member band from a pool of 5 guitarists, 4 drummers, and 3 keyboardists, all of whom are distinct individuals. The band must include at least one drummer, must include strictly more guitarists than drummers, and may include at most one keyboardist. How many different 5-member bands can the director form?
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Correct answer
D
The keyboardist count is capped at one, so split on it; the strictly-more-guitarists-than-drummers and at-least-one-drummer conditions then fix the allowed splits.
Zero keyboardists: the 5 members come from guitarists and drummers with more guitarists than drummers and at least one drummer. The only splits are 4 guitarists with 1 drummer, C(5,4) × C(4,1) = 5 × 4 = 20, and 3 guitarists with 2 drummers, C(5,3) × C(4,2) = 10 × 6 = 60. That case totals 80.
Exactly one keyboardist: choose the keyboardist in C(3,1) = 3 ways, then fill 4 seats from guitarists and drummers under the same conditions. The only valid split is 3 guitarists with 1 drummer, since a 4-and-0 split has no drummer and a 2-and-2 split is not strictly more: C(5,3) × C(4,1) = 10 × 4 = 40, × 3 = 120.
Total: 80 + 120 = 200. So the answer is D.
Three interacting constraints mean no single combination works, so the count must be assembled case by case. A (80) keeps only the no-keyboardist case. B (120) reads at most one as exactly one and keeps only that case. C (180) drops the 4-guitarists-1-drummer lineups. E (320) ignores the keyboardist cap and allows any number.
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