Question

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?

$12$

$16$

$28$

$32$

$40$

Answer: C

Concepts:arrangements with restrictions inclusion-exclusion

Difficulty rating: 1730

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.

2017 AMC 12A Problem 14

Answer: C

Difficulty rating: 1730

Solution:

Problem 14 in Other Years