Multiple choice

How many words can be formed from the letters of the word $'AFTER'$ so that the vowels never come together?

  1. $72$
  2. $120$
  3. $48$
  4. $52$
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A Correct answer
Explanation

Total letters in AFTER = 5. Vowels = A, E. Consonants = F, T, R. To keep vowels apart, arrange consonants first: 3! = 6 ways. There are 4 gaps between consonants: _ F _ T _ R _. Place 2 vowels in 4 gaps: 4P2 = 4 * 3 = 12. Total = 6 * 12 = 72.

AI explanation

The word AFTER has 5 letters, so its total arrangements equal 5 factorial, which is 120. To find the arrangements where the 2 vowels (A, E) never come together, we subtract the arrangements where they are together from the total. Treating the 2 vowels as one unit creates 4 units, which arrange in 4 factorial ways, and the vowels arrange in 2 factorial ways, giving 24 multiplied by 2 for a total of 48 together arrangements. Subtracting 48 from 120 leaves 72 arrangements where the vowels do not come together.

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