Ways to arrange seats where a certain group of people has to be together.

Question

I just had this question on one of my quizzes and I have tried to do the question.

So, we have 4 types of dogs on the day of the photoshoot, 1 golden retriever, 1 german shepherd, 4 chihuahuas and 4 Dachshund.

There are 10 seats at the photoshoot.

Q1 - How many ways to arrange the seat for the dogs

10 seats = 10! = 3628800

Q2 - How many ways to arrange the seat if the dachshunds need to sit together (Ex. xxDDDDxxxx)

Counting 4 dachshunds as 1 item, [DDDD]xxxxxx, therefore there are 7! = 5040 arrangements where the dachshunds sit together.

Maybe it is the wording of the question that is bugging me that something is not right.

If someone could check this for me and provide further examples that would be really helpful

Answer 1

In Q1 and the non-dachshund part of Q2, you're counting the dogs as all being distinguishable. That suggests you should also factor in the order of the four dachshunds in the DDDD block. Otherwise your work looks good.

Answer 2

You were on the right track, but the 4 dachshunds can also be permuted. This gives you $4!7!$.