How many words can be formed using the letter | vedclass

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(D) The word $INSURANCE$ consists of $9$ letters: $I, N, S, U, R, A, N, C, E$.The vowels are $I, U, A, E$ ($4$ vowels).The consonants are $N, S, R, N,

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) The word $INSURANCE$ consists of $9$ letters: $I, N, S, U, R, A, N, C, E$.The vowels are $I, U, A, E$ ($4$ vowels).The consonants are $N, S, R, N, C$ ($5$ consonants).Since all vowels must come together,we treat the group $(I, U, A, E)$ as a single unit.Now,we have the unit $(IUAE)$ and the $5$ consonants $(N, S, R, N, C)$,making a total of $6$ units.These $6$ units can be arranged in $\frac{6!}{2!}$ ways (since $N$ repeats twice).Within the vowel unit,the $4$ vowels can be arranged in $4!$ ways.Total number of words = $\frac{6!}{2!} 	imes 4! = \frac{720}{2} 	imes 24 = 360 	imes 24 = 8640$.Since $8640$ is not among the options,the correct choice is $D$.

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

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