Multiple choice

DIRECTIONS: Type in your answer in the input box provided below the question. In how many ways can the letters of the word NATION be arranged such that no two vowels occupy adjacent positions?

  1. 72

  2. 70

  3. 68

  4. 60
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

NATION has 6 letters: N, A, T, I, O, N. Vowels are A, I, O (3). Consonants are N, T, N (3). Arrange consonants first: 3!/2! = 3 ways (N, T, N). There are 4 gaps between consonants: _ C _ C _ C _. Place 3 vowels in 4 gaps: 4P3 = 4*3*2 = 24. Total = 3 * 24 = 72.

AI explanation

The word NATION has 6 letters with vowels (A, I, O) and consonants (N, N, T). First arrange the 3 consonants, which can be done in 3! / 2! = 3 ways. This creates 4 gaps around the consonants where the 3 distinct vowels can be placed to ensure no two vowels are adjacent. Selecting and arranging the vowels in these gaps gives 4P3 = 4 * 3 * 2 = 24 ways. Multiplying the consonant arrangements by the vowel arrangements gives 3 * 24 = 72 ways.

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