Multiple choice

Find the number of different words that can be formed from the letters of the word TRIANGLE, so that no vowels are together.

  1. $14000$
  2. $14500$
  3. $14400$
  4. $14402$
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C Correct answer
Explanation

The word TRIANGLE has 8 distinct letters, 3 vowels (A, I, E) and 5 consonants (T, R, N, G, L). To keep vowels apart, arrange the 5 consonants first (5! = 120 ways) and place the 3 vowels in the 6 available gaps (6P3 = 6 * 5 * 4 = 120 ways). Total = 120 * 120 = 14400.

AI explanation

The word TRIANGLE has 8 distinct letters, comprising 5 consonants and 3 vowels. Arrange the 5 consonants first, which can be done in 5 factorial, or 120, ways. This arrangement creates 6 gaps (including the ends) where the vowels can be placed to ensure they are not together. Select and arrange the 3 vowels in these 6 gaps using the permutation method, giving 6P3, which equals 120. The total number of words is 120 multiplied by 120, which equals 14400.

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