Multiple choice

In how many distinct ways can the letters of the word "TRIANGLE" be arranged such that the vowels are placed only in positions that are prime numbers?

  1. 2,880

  2. 880

  3. 280

  4. 80
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A Correct answer
Explanation

The word TRIANGLE has 8 letters: 3 vowels (I, A, E) and 5 consonants (T, R, N, G, L). Prime positions are 2, 3, 5, 7. There are 4 prime positions for 3 vowels. Ways to choose and arrange vowels = 4P3 = 4*3*2 = 24. Ways to arrange 5 consonants in remaining 5 positions = 5! = 120. Total ways = 24 * 120 = 2880.

AI explanation

The word TRIANGLE has 8 positions, of which four are prime (2, 3, 5, and 7). The three vowels can be placed in these four prime positions in 4P3 ways, which equals 24. The five consonants can be arranged in the remaining five positions in 5! ways, which is 120. Multiplying these gives 24 multiplied by 120, yielding 2880 arrangements.

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