There are four married couples and eight seats. When they sit, what is the probability that husband sits adjacent to his wife?
The answer to this problem is 12/35
I can arrive at this answer when I try to calculate the probability of the complement and then subtract it from 1. i.e. 1 - Probability that one couple sit together - Probability that two couples sit together - Probability that three couples sit together - Probability that four coupls sit together
However when I try to calculate the probability straight, I can't do it right.
This is how I'm doing it:
Denominator will be 8! Building the numerator:
Numerator will be a product. The first seat can be occupied by 8 people, the second seat can be occupied by 7 people, but one of them could be the first seated person's spouse, so second term will be 6. Numerator so far: 8 x 6
Third seat can be occupied by five people but one of the could be the spouse of the person on the second seat, so third term will be 5. Numerator so far: 8 x 6 x 5
Similarly we get: 8 x 6 x 5 x 4 x 3 x 3 x 2
(Note that on the seven seat we get the number three again because we are sure that this person's spouse has been seated in one of the first four seats)
But by these calculations I get the answer 3/7
What am I doing wrong?