Victoria refuses to sit next to either William or Xavier. Yasmin refuses to sit next to Zack. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
Let Victoria be V, William be W, Xavier be X, Yasmin be Y and Zack be Z.
Scenario 1: V sits on an end seat.
Then, since W and X can’t sit next to V, that must mean either Y or Z sits next to V.
After picking either Y or Z, then either W or X must sit next to Y/Z.
Then, the last two people can be arranged in two ways.
Since there are two different end seats that V can sit in, there are a total of .
Scenario 2: V does not sit in an end seat.
The are 3 seats for V to choose from
In this case, then only two people that can sit next to V are Y and Z, and they
can be in either order and there are two ways to arrange W and X
So the total is 28
Adapted from http://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_19