Multiple choice

How many words can be formed from the letters of the word $'SIGNATURE'$ so that all the vowels come together?

  1. $17280$
  2. $4320$
  3. $720$
  4. $80$
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A Correct answer
Explanation

SIGNATURE has 9 letters: S, G, N, T, R (consonants) and I, A, U, E (vowels). Treat the 4 vowels as one block. We have 5 consonants + 1 block = 6 items. They can be arranged in 6! ways. The 4 vowels can be arranged among themselves in 4! ways. Total = 6! * 4! = 720 * 24 = 17280.

AI explanation

The word SIGNATURE has 9 distinct letters, consisting of 4 vowels and 5 consonants. Treating the 4 vowels as a single connected unit gives 6 total units to arrange, which can be done in 6! ways. The 4 vowels within the unit can be arranged among themselves in 4! ways, so multiplying 720 by 24 yields 17280 total arrangements.

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