Final answer:
The number of arrangements of the word TRIANGLE that ensures no two vowels are adjacent is 14400, calculated by arranging consonants, selecting spaces for vowels, and permuting the vowels.
Step-by-step explanation:
The question is about finding the number of ways in which the letters of the word TRIANGLE can be arranged such that no two vowels are adjacent. We first identify the vowels in the word TRIANGLE, which are A, E, and I. There are 5 consonants - T, R, N, G, L. The consonants can be arranged in 5! ways, which is 120 ways.
Each arrangement of consonants gives us 6 spaces (before the first consonant, between the consonants, and after the last consonant) to place the vowels such that no two vowels are adjacent. Since we have 3 vowels and 6 spaces, we can select 3 spaces out of the 6 to place the vowels in C(6,3) ways which are 20 ways. The vowels themselves can be arranged among themselves in 3! which is 6 ways. Therefore, the total number of arrangements where no vowels are adjacent is 120 (arrangements of consonants) × 20 (ways to choose the spaces for vowels) × 6 (arrangements of vowels), which equals 14400