In how many ways can the letters of the word | vedclass

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(A) The word $PERMUTATIONS$ contains $12$ letters: $P, E, R, M, U, T, A, T, I, O, N, S$.The vowels are $E, U, A, I, O$ ($5$ vowels,all distinct).The c

Vedclass · Accepted Answer

$2419200$

Vedclass · Answer

(A) The word $PERMUTATIONS$ contains $12$ letters: $P, E, R, M, U, T, A, T, I, O, N, S$.The vowels are $E, U, A, I, O$ ($5$ vowels,all distinct).The consonants are $P, R, M, T, T, N, S$ ($7$ consonants,with $T$ repeating twice).Since the $5$ vowels must be together,we treat them as a single unit. This unit plus the $7$ consonants gives us $8$ objects to arrange.These $8$ objects contain $2$ $T$s,so the number of arrangements is $\frac{8!}{2!}$.The $5$ distinct vowels within their unit can be arranged in $5!$ ways.Total arrangements $= \frac{8!}{2!} 	imes 5! = \frac{40320}{2} 	imes 120 = 20160 	imes 120 = 2419200$.

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if the vowels are all together?

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