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

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Question

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

Vedclass · Accepted Answer

$14400$

Vedclass · Answer

(C) The word $TRIANGLE$ contains $8$ distinct letters: $T, R, I, A, N, G, L, E$.There are $5$ consonants $(T, R, N, G, L)$ and $3$ vowels $(I, A, E)$.First,arrange the $5$ consonants in $5! = 120$ ways.These $5$ consonants create $6$ gaps (including the ends) where the $3$ vowels can be placed so that no two vowels are together: $\_ C \_ C \_ C \_ C \_ C \_$.The number of ways to choose and arrange $3$ vowels in these $6$ gaps is $^6P_3 = 6 	imes 5 	imes 4 = 120$.Therefore,the total number of arrangements is $120 	imes 120 = 14400$.

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

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