PrepLattice is an independent test-preparation service and is not affiliated with or endorsed by GMAC, the organization that administers the GMAT. GMAT and GMAT Focus are trademarks of GMAC, used here only to name the exam this question is designed to prepare you for.
A 5-member panel is to be chosen from 6 economists and 4 historians and then seated in a row, subject to the rule that no 2 historians may sit next to each other. The panel must contain exactly 2 historians and 3 economists. In how many ways can the panel be both chosen and seated?
Five fresh questions every day, your progress tracked, every miss explained. Free with an account.
Correct answer
C
First select the members: choose 3 economists from 6 and 2 historians from 4, C(6,3) × C(4,2) = 20 × 6 = 120 selections. Now seat 3 economists and 2 historians in a row with no two historians adjacent.
Seat the 3 economists first in 3! = 6 orders; they create 4 gaps (before, between, between, after) into which the 2 historians go in separate gaps, and the historians are placed and ordered in P(4,2) = 4 × 3 = 12 ways. Seatings per selection = 6 × 12 = 72.
Total = 120 × 72 = 8640. Verification by total-minus-adjacent: total seatings of 5 distinct people = 5! = 120; seatings with the 2 historians adjacent (glue them) = 4! × 2! = 24 × 2 = 48; valid = 120 − 48 = 72, and 120 × 72 = 8640 confirms the gap count. Choice (D) 14400 uses all 5! seatings, and (B) 5760 uses the forbidden adjacent seatings.
Pick the theme that suits how you work. You can change it anytime.