permutation, arrangement with restrictions, playing with vowels

Question

in how many ways can the letters of the word "together" be arranged? In how many of these arrangements are all the vowels together?

i solver the first one: (8P8)/(2P2 *2P2) =10080

but not sure with the vowels

Answer 1

I concur with the first one: $\frac{8!}{2!2!} = 10080$, as there are $8!$ many permutations, and we can interchange both t's and both e's, so we get 4 times fewer.

Well, for the other one: there are 3 vowels that have to be together. Call all vowels together "V". Then we have to permute tgthrV (so 6 symbols) (where V stands for all the vowels). We have two t's so we get $\frac{6!}{2!} =360$ different permuations of tgthrV.

But we have to replace V in every such permutation by a permutation of the actual vowels, of which there are 3 different ones (same formula: $\frac{3!}{2!} = 3$, or enumerate oee, eoe, eeo).

So we get 3 times as many, so $3 \times 360 = 1080$.