Combinatorics question about six letter sequences with repetition

The question I'm trying to answer is as follows:

"How many six-letter “words” (sequences of letters with repetition) are there in which the first and last letter are vowels? In which vowels appear only (if at all) as the first and last letter?"

For the first part of the problem, I got an answer of $5\cdot 26\cdot26\cdot26\cdot26\cdot5 = 11424400$.

I'm not sure this is correct because I'm not sure if some solutions are being double counted.

For the second part, I'm having trouble finding an answer. For solutions with the vowels, I think it would be $5\cdot21\cdot21\cdot21\cdot21\cdot5$, but then I'm not sure how to account for those solutions that do not have vowels.

Any help is appreciated. Thank you!