Problem:
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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the set of ways to seat the five people in which Alice sits next to Bob. Let be the set of ways to seat the five people in which Alice sits next to Carla. Let be the set of ways to seat the five people in which Derek sits next to Eric. The required answer is . The Inclusion-Exclusion Principle gives
.
Viewing Alice and Bob as a unit in which either can sit on the other's left side shows that there are elements of . Similarly there are 48 elements of and 48 elements of . Viewing Alice, Bob, and Carla as a unit with Alice in the middle shows that . Viewing Alice and Bob as a unit and Derek and Eric as a unit shows that . Similarly . Finally, there are elements of . Therefore , and the answer is .
OR
There are three cases based on where Alice is seated.
If Alice takes the first or last chair, then Derek or Eric must be seated next to her, Bob or Carla must then take the middle chair, and either of the remaining two individuals can be seated in either of the other two chairs. This gives a total of arrangements.
If Alice is seated in the second or fourth chair, then Derek and Eric will take the seats on her two sides, and this can be done in two ways. Bob and Carla can be seated in the two remaining chairs in two ways, which yields a total of arrangements.
If Alice sits in the middle chair, then Derek and Eric will be seated on her two sides, with Bob and Carla seated in the first and last chairs. This results in arrangements.
Thus there are possible arrangements in total.
The problems on this page are the property of the MAA's American Mathematics Competitions