In how many ways can the letters of the word | vedclass

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

Vedclass · Accepted Answer

$14400$

Vedclass · Answer

(C) The word $TRIANGLE$ has $8$ letters: $T, R, I, A, N, G, L, E$.There are $3$ vowels $(I, A, E)$ and $5$ consonants $(T, R, N, G, L)$.First,arrange the $5$ consonants in $5!$ ways: $5! = 120$.These $5$ consonants create $6$ gaps (including ends) where the $3$ vowels can be placed: $\_ C \_ C \_ C \_ C \_ C \_$.The number of ways to choose and arrange $3$ vowels in $6$ gaps is $^6P_3 = 6 	imes 5 	imes 4 = 120$.Total arrangements = $120 	imes 120 = 14400$.

In how many ways can the letters of the word $TRIANGLE$ be arranged such that no two vowels occur together?

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