Multiple choice

Consider the letters of the word MATHEMATICS. Possible number of words in which no two vowels are together is

  1. $7!^8C_4\dfrac {4!}{2!}$
  2. $\dfrac {7!}{2!}^8C_4\dfrac {4!}{2!}$
  3. $\dfrac {7!}{2!2!}^8C_4\dfrac {4!}{2!}$
  4. $\dfrac {7!}{2!2!2!}^8C_4\dfrac {4!}{2!}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

MATHEMATICS has 11 letters: M(2), A(2), T(2), H, E, I, C, S. Vowels: A, A, E, I (4). Consonants: M, M, T, T, H, C, S (7). Arrange 7 consonants: 7!/(2!2!). There are 8 gaps for 4 vowels. Choose 4 gaps: 8C4. Arrange 4 vowels: 4!/2!. Total = (7!/(2!2!)) * 8C4 * (4!/2!).

AI explanation

The word MATHEMATICS has 7 consonants (M, T, H, M, T, C, S) and 4 vowels (A, A, E, I). The consonants can be arranged in 7! / (2! 2!) ways, creating 8 gaps for the vowels. The 4 vowels are chosen and placed in these gaps using 8C4, and they are arranged in 4! / 2! ways. Multiplying these values gives the total arrangements as (7! / (2! 2!)) * 8C4 * (4! / 2!). The correct result is (7! / (2! 2!)) * 8C4 * (4! / 2!).

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