Using all the letters of the word ARTFUL, how many words start with a vowel but end with a consonant?

The word ARTFUL has two vowels – A and U.
If a word starts and ends with a vowel, the two letters to occupy the first and the last positions can be selected and arranged in 2P2 = 2 × 1 = 2 ways.
The remaining 4 letters can be arranged among themselves in 4P4 = 4! = 24 ways.
The number of words that start and end with a vowel = 2 × 24 = 48.

If a word starts with a vowel but ends with a consonant, its first letter can be selected from A and U = 2 ways.
Its last letter can be selected from R, T, F, L = 4 ways.
The remaining 4 letters can be arranged in 4! = 24 ways.