Multiple choice

How many seven lettered words can be formed with the letters of the word $'MISTAKE'$ such that no two vowels come together?

  1. $7!$
  2. $4!\times3!$
  3. $4!\times ^5P_3$
  4. $5!\times ^4P_2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

MISTAKE has 7 letters: 3 vowels (I, A, E) and 4 consonants (M, S, T, K). To keep vowels apart, place consonants first: _ C _ C _ C _ C _. There are 5 spaces for 3 vowels. Ways = 4! * 5P3 = 24 * 60 = 1440.

AI explanation

The word MISTAKE has 4 consonants and 3 distinct vowels, and to ensure no two vowels are together, first arrange the 4 consonants in 4! ways. These consonants create 5 gaps (including the ends), and the 3 vowels must be placed in 3 of these 5 gaps, which is done using the permutation formula 5P3. The total number of valid arrangements is the product of these two steps, giving 4! * 5P3.

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