Multiple choice

How many words can be formed by using the letters of the word ' INSURANCE ',when vowels always remain together ?

  1. $8640$
  2. $17280$
  3. $720$
  4. $None$ $of$ $these$
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A Correct answer
Explanation

The word INSURANCE has 9 letters: I, N, S, U, R, A, N, C, E. Vowels are I, U, A, E (4). Consonants are N, S, R, N, C (5). Treating the 4 vowels as one block, we arrange 6 items (the block + 5 consonants) in 6!/2! ways (due to two Ns). The 4 vowels can be arranged in 4! ways. Total = (6!/2!) * 4! = 360 * 24 = 8640.

AI explanation

The word INSURANCE has 9 letters with repeated N (2 times), while the 4 vowels are I, U, A, E. Considering the 4 vowels as a single unit, we have 6 units to arrange (5 consonants plus the vowel block), which gives 6! divided by 2! arrangements, equaling 360. The 4 vowels within their block can be arranged in 4! ways, which is 24. Multiplying these gives 360 times 24, so the result is 8640.

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