How many words can be formed using the letter | vedclass

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(A) The word $INDEPENDENCE$ contains $12$ letters: $I, N, D, E, P, E, N, D, E, N, C, E$.The vowels are $I, E, E, E, E$ ($4$ $E$'s and $1$ $I$).The con

Vedclass · Accepted Answer

$16800$

Vedclass · Answer

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

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

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