In how many different ways, can the letters of the words 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 $$2!=2$$ ways.

Number of arrangements having vowels together = $$(24 \times 2) = 48$$ ways

Total arrangements of all letters = $$5!=120$$ ways

=> Number of arrangements not having vowels together = $$(120 - 48) = 72$$ ways

=> Ans - (D)