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:

#1color(white)(0)2color(white)(0)3color(white)(0)4color(white)(0)5color(white)(0)6color(white)(0)7color(white)(0)8color(white)(0)9color(white)(0)10#

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 #(3!)=6# ways, so the sisters can sit together #8xx6=48# ways.

Now we have 7 seats left and 3 people to fill them. Again, we'll look at this as a Permutation:

#P_(7,3)=(7!)/((7-3)!)=(7!)/(4!)=(7xx6xx5xx4!)/(4!)=210# ways for each of the 48 ways the sisters can be seated.

And so there are #48xx210=10,080# ways they can be seated.