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+1 vote

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

$\begin{array}{1 1}(A)\;14300\\(B)\;14200\\(C)\;14400\\(D)\;14500\end{array}$

Toolbox:
• $P(n,r)=\large\frac{n!}{(n-r)!}$
• $n!=n(n-1)(n-2)(n-3).....(3)(2)(1)$
Given
TRIANGLE
Total no of vowels =3
Total no of consonants =5
The vowels can be placed in $\rightarrow 6P_3$
$\Rightarrow \large\frac{6!}{3!}$
$\Rightarrow \large\frac{6\times 5 \times 4\times 3!}{3!}$
$\Rightarrow 120$ ways
The consonants can be placed in their places in 5!
$\Rightarrow 5\times 4\times 3\times 2\times 1$
$\Rightarrow 120$ways
Total no of ways =$120\times 120$
$\Rightarrow 14400$ ways
Hence (C) is the correct answer.