"How many permutations exist of the letters in the word COMPUTER?"

I said8!because there are 8 letters and so it should just be n! where n is the number of choices. Right?

"How many permutations end in a vowel?"

I thought7!3!but I'm not certain. Because there are 3 vowels and I want one of them in the last position, and the other 7 letters can be whatever else is left..

Is this correct? If not, can someone tell me what I am doing wrong?