In how many different ways can the letters of | vedclass

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(D) The word '$CORPORATION$' consists of $11$ letters: $C, O, R, P, O, R, A, T, I, O, N$.The vowels are $O, O, A, I, O$ (total $5$ vowels,where $O$ re

Vedclass · Accepted Answer

$50400$

Vedclass · Answer

(D) The word '$CORPORATION$' consists of $11$ letters: $C, O, R, P, O, R, A, T, I, O, N$.The vowels are $O, O, A, I, O$ (total $5$ vowels,where $O$ repeats $3$ times).The consonants are $C, R, P, R, T, N$ (total $6$ consonants,where $R$ repeats $2$ times).Treating the $5$ vowels as a single unit,we have $6$ consonants + $1$ unit of vowels = $7$ items to arrange.The number of ways to arrange these $7$ items,where $R$ repeats $2$ times,is $\frac{7!}{2!} = \frac{5040}{2} = 2520$.Within the vowel unit,the $5$ vowels $(O, O, O, A, I)$ can be arranged in $\frac{5!}{3!} = \frac{120}{6} = 20$ ways.Total number of arrangements = $2520 	imes 20 = 50400$.

In how many different ways can the letters of the word '$CORPORATION$' be arranged so that the vowels always come together?

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