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Two couples and one single person are seated at random in a [#permalink]
10 Oct 2004, 14:31

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?

I like all the ways given, but not sure someone without a complete comprehension of probability and permutations would get it. I have a way that's a little more cumbersome, but it works:

There are 5 spaces to fill up:
_ _ _ _ _

The denominator here will be 120, since that's the total number of ways 5 people could sit in 5 seats.

The number of permutations for the numerator would be this:

the single person could be in any of those 5 spaces:

S _ _ _ _
_ S _ _ _
_ _ S _ _
_ _ _ S _
_ _ _ _ S

Now go through and think logically about how many people could fill each of the other spaces:

S 4 2 1 1 = 8
4 S 2 1 1 = 8
4 2 S 2 1 = 16
4 2 1 S 1 = 8
4 2 1 1 S = 8

So the total number of possibilities is 48/120 = 2/5

I know this seems longer, but some people (like myself) think more clearly this way, rather than trying to conceptualize the bouncing around of couples in a permutations world. Since there are only 120 total possibilities, it's not impossible to see this method being worthwhile on the test, even if it adds an extra minute.