Multiple choice

How many words can be made out from the letters of the word $INDEPENDENCE$, in which vowels always come together?

  1. $16800$
  2. $16630$
  3. $1663200$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In the word INDEPENDENCE, there are 12 letters: 7 consonants (3 N's, 2 D's, 1 P, 1 C) and 5 vowels (4 E's, 1 I). Treating the 5 vowels as a single block, we have 8 units to arrange, which can be done in 8! / (3! * 2!) = 3,360 ways. The 5 vowels within their block can be arranged in 5! / 4! = 5 ways, giving a total of 3,360 * 5 = 16,800 arrangements.

AI explanation

The word INDEPENDENCE has 5 vowels (4 E's and 1 I) and 7 consonants (3 N's, 2 D's, and 2 P's). Treating all the vowels as a single block, we have 8 total items to arrange, which yields 8!/(3!2!2!) ways. The internal arrangement of the vowels within the block can be done in 5!/4! ways. Multiplying these gives (8!/(3!2!2!)) x (5!/4!) = 16800.

Keep practicing — related questions