I have a question about combinatorics.
Here is the question:
A waiting area outside the doctor's office contains a row of 7 chairs. In how many different ways can a man, a woman and a boy occupy 3 of the chairs such that:
a. the man and the boy seated in adjoining chairs
b. all three seated in adjoining chairs
a. Since the man and the boy must be seated in adjoining chairs, so we can conclude that only 2 groups sits in 3 chairs here, so the number of ways is:
P(6,2) * 2= 60 ways
b. Because three people must be seated near to each others, so we can group them such that:
number of ways of sitting: P(5,1)
Because there is 3 people, so the total number of ways of sitting is: 3*P(5,1)= 15 ways
After that, I doubt about my answers. Are my solutions is true or I need to improve that?