March 2017 Brain Puzzler Solution

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

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