Question

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

Answer 1

Total number of letters in the word TRIANGLE $=8$

Out of $8$ letters, $5$ are consonants and $3$ are vowels

If vowels are not to be together, we have the following arrangements

$VCVCVCVCVC$

Consonants can be arranged in $5!=120$ ways

Vowels occupy $6$ places

(gaps between consonants)

$3$ vowels can be arranged in $6$ places $={}^6{P}_3$

$=\frac{6!}{(6-3)!}=120$ ways

Total arrangements $=120\times 120$
$=14400$ ways

$\therefore$ Number of different words that can be formed $=14400$