Multiple choice

The number of arrangements that can be formed out of 'LOGARITHM' so that no two vowels come together is

  1. $6!\ ^7P_3$
  2. $6!\ 7!$
  3. $6!\ 3!$
  4. $7!\ 3!$
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A Correct answer
Explanation

LOGARITHM has 9 letters: 3 vowels (O, A, I) and 6 consonants (L, G, R, T, H, M). To keep vowels apart, place consonants first: 6! ways. This creates 7 gaps. Place 3 vowels in 7 gaps: 7P3 ways. Total = 6! * 7P3.

AI explanation

The word LOGARITHM has 9 distinct letters including the 3 vowels O, A, and I, and the 6 consonants L, G, R, T, H, and M. To keep the vowels separate, first arrange the 6 consonants in 6! ways, which creates 7 available spaces around and between them. We then select and arrange the 3 distinct vowels in these 7 spaces using the permutation formula 7P3, giving a total of 6! 7P3 arrangements.

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