Question

The number of ways in which the letters of the word $\mathrm{TRIANGLE}$ can be arranged such that two vowels do not occur together is?

• A. $1200$
• B. $2400$
• C. $14400$
• D. None of these

Answer 1

The correct option is C

$14400$

The explanation for the correct answer.

Find the number of ways such that no two vowels occur together.

Given: $\mathrm{TRIANGLE}$

The total number of letters in the word triangle is $8$.

The total number of words that can be formed $=8!=40320$.

The total number of words in which two vowels occur together $={}^3{C}_2\times 7!\times 2!=30240$.

The number of ways where all vowels occur together $={}^3{\mathrm{C}}_3\times 6!\times 3!=4320$.

Therefore required number of words $=40320-30240+4320=14400$.

Hence option $\left(C\right)$ is the correct answer.