In a waiting room are 10 seats. Three sisters want to sit together and there are three other people as well. In how many ways can the 6 people sit in the waiting room?
Before we move forward with this question, I'm going to assume that the seats are all in a row (although what waiting room has 10 seats in a row is a question for another day...). I'm also going to assume that each person is an individual and so having one sister or another sister sit in a seat makes a difference (vs simply having a sister sit in a seat).
Let's do this with a bit of a visual. First off we have 10 seats:
Ok - now first let's talk about the 3 sisters: (A)bby, (B)arb, (C)arol. They want to sit together - so that means they can sit in seats: (1,2,3), (2,3,4), (3,4,5), etc - in all, there are 8 groups of seats they can take up.
Within that group of seats, the sisters can sit
Now we have 7 seats left and 3 people to fill them. Again, we'll look at this as a Permutation:
And so there are