The number of ways in which the letters of th | vedclass

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Question

(C) The word $TRIANGLE$ consists of $8$ distinct letters: $T, R, I, A, N, G, L, E$.
There are $3$ vowels $(I, A, E)$ and $5$ consonants $(T, R, N, G,

Vedclass · Accepted Answer

$14400$

Vedclass · Answer

(C) The word $TRIANGLE$ consists of $8$ distinct letters: $T, R, I, A, N, G, L, E$.
There are $3$ vowels $(I, A, E)$ and $5$ consonants $(T, R, N, G, L)$.
To ensure no two vowels occur together, we first arrange the $5$ consonants in $5!$ ways:
$5! = 120$
These $5$ consonants create $6$ gaps:
$. C . C . C . C . C .$
Now we select $3$ gaps out of $6$ for placing the vowels:
$^6C.3 = 20$
The $3$ vowels can be arranged in these selected gaps in:
$3! = 6$ ways
Therefore, total arrangements:
$5! 	imes {}^6C.3 	imes 3!$
$= 120 	imes 20 	imes 6$
$= 14400$

The number of ways in which the letters of the word $TRIANGLE$ can be arranged such that no two vowels occur together is

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