Problem Solving

Alright, so let's go ahead and refer to these folks as A, B, C, D, and E.

We have 5 total people. This means they can be arranged in 5! = 120 ways. However, some of these arrangements are not permitted.

We know that A and D cannot be next to each other. This means that A and D cannot be in 8 different positions:

AD _ _ _
DA _ _ _
_ AD _ _
_ DA _ _
_ _ AD _
_ _ DA _
_ _ _ AD
_ _ _DA

For each of those positions, B, C, and E can be arranged in 3! = 6different ways: BCE, BEC, CBE, CEB, EBC, and ECB. So there are 8 * 3! = 8 * 6 = 48 arrangements that are not permitted. This means that 120 - 48 = 72 arrangements are permitted, giving us D as the correct answer.

A good strategy with complex combination problems is often to start with your total possible number of arrangements and then take away the ones that aren't permitted one step at a time - this often goes faster than trying to find the ones that are permitted.

lheiannie07 wrote:Adam, Bob, Carol, Diane, and Ed are all sitting on a bench. If Adam and Diane cannot sit next to each other, how many different seating arrangements are possible on the bench?

A. 5
B. 10
C. 48
D. 72
E. 120

We can use the following equation:

The number of ways Diane is not next to Adam = total number of ways to arrange the group - number of ways Diane is next to Adam

The total number of ways to arrange the group is 5! = 120.

One way for Diane to sit next to Adam is:

[A-D][C][D]

Notice that we have made Adam and Diane one unit, so we now have 4 total positions on the bench that can be arranged in 4! = 24 ways. However, there are also 2! = 2 ways to arrange Adam and Diane, so the total number of ways to arrange the group with Diane next to Adam is 24 x 2 = 48. Thus:

The number of ways Diane is not next to Adam = 120 - 48 = 72.

Answer: D