took me almost forever to get why you have 16 for this _ _ 5 _ _, whew!

That was pure madness.

marcodonzelli wrote:

Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

Choices are 1/5, 1/4 , 3/8, 2/5 and 1/2

There are 5! = 120 ways to seat the 5 people.

I'll use the same naming of the people. (1,2) (3,4) and 5 as the single.

5 _ _ _ _ = _ _ _ _ 5 Now how many ways can we sit people without couples touching?

4*2*1*1 = 8

and since 5 _ _ _ _ = _ _ _ _ 5 we have 8*2 = 16 ways to seat people without couples touching when the single is on the end.

_ 5 _ _ _ = _ _ _ 5 _ Now how many ways can we sit people without couples touching?

4*2*1*1 = 8

and since we have two seating arranges for that to work 8*2 = 16 ways to seat people when the single is sitting one spot from the end.

_ _ 5 _ _ with no mirror image

4*2*2*1 = 16

16+16+16 = 48

48/120 = 2/5

Answer 2/5