2017 AMC 12A Problem 14

Below is the professionally curated solution for Problem 14 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:arrangements with restrictions inclusion-exclusion

Difficulty rating: 1730

14.

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?

$12$

$16$

$28$

$32$

$40$

Solution:

Let $X,$ $Y,$ $Z$ be the seatings where Alice-Bob, Alice-Carla, and Derek-Eric are adjacent, respectively. The answer is $5!-|X\cup Y\cup Z|.$

By inclusion-exclusion, $|X\cup Y\cup Z|=(48\cdot3)-(12+24+24)+8=92,$ so the answer is $120-92=28.$

Thus, the correct answer is C.

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