How many words can be formed using the letter | vedclass

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(A) The word $INDEPENDENCE$ consists of $12$ letters: $I, N, D, E, P, E, N, D, E, N, C, E$.The vowels are $I, E, E, E, E$ (total $5$ vowels).The conso

Vedclass · Accepted Answer

$16800$

Vedclass · Answer

(A) The word $INDEPENDENCE$ consists of $12$ letters: $I, N, D, E, P, E, N, D, E, N, C, E$.The vowels are $I, E, E, E, E$ (total $5$ vowels).The consonants are $N, D, P, N, D, N, C$ (total $7$ consonants).Since all vowels must come together,we treat the group of $5$ vowels $(I, E, E, E, E)$ as a single unit.Now,we have $7$ consonants + $1$ unit of vowels = $8$ entities to arrange.The number of ways to arrange these $8$ entities,where $N$ appears $3$ times and $D$ appears $2$ times,is $\frac{8!}{3! 	imes 2!}$.Within the vowel unit,the $5$ vowels $(I, E, E, E, E)$ can be arranged in $\frac{5!}{4!}$ ways (since $E$ repeats $4$ times).Total number of words = $\frac{8!}{3! 	imes 2!} 	imes \frac{5!}{4!} = \frac{40320}{6 	imes 2} 	imes 5 = \frac{40320}{12} 	imes 5 = 3360 	imes 5 = 16800$.

How many words can be formed using the letters of the word $INDEPENDENCE$,such that all vowels always come together?

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