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.
