In how many different ways can the letters of the word EXTRA be arranged so that the vowels are never together?

Taking the vowels (EA) as one letter, the given word has the letters XTR (EA), i.e., 4 letters.
These letters can be arranged in 4! = 24 ways
The letters EA may be arranged amongst themselves in 2 ways.
Number of arrangements having vowels together = (24 × 2) = 48 ways
Total arrangements of all letters
= 5!
= (5 × 4 × 3 × 2 × 1)
= 120
Number of arrangements not having vowels together
= (120 - 48)
= 72