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

$8640$

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 entity.Now,we have the set ${ (IUAE), N, S, R, N, C }$,which contains $6$ entities.In this set,the letter $N$ repeats $2$ times.The number of ways to arrange these $6$ entities is $\frac{6!}{2!} = \frac{720}{2} = 360$.Within the vowel group $(I, U, A, E)$,there are $4$ distinct vowels,which can be arranged in $4! = 24$ ways.Total number of words = $360 	imes 24 = 8640$.

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

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