In how many different ways can the letters of | vedclass

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(D) The word '$AUCTION$' consists of $7$ distinct letters: $A, U, C, T, I, O, N$.There are $4$ vowels $(A, U, I, O)$ and $3$ consonants $(C, T, N)$.To

Vedclass · Accepted Answer

$576$

Vedclass · Answer

(D) The word '$AUCTION$' consists of $7$ distinct letters: $A, U, C, T, I, O, N$.There are $4$ vowels $(A, U, I, O)$ and $3$ consonants $(C, T, N)$.To ensure the vowels always come together,we treat the group of $4$ vowels $(A, U, I, O)$ as a single entity or 'block'.Now,we have the block of $4$ vowels and the $3$ consonants,making a total of $4$ entities to arrange: $(AUIO), C, T, N$.These $4$ entities can be arranged in $4!$ ways.$4! = 4 	imes 3 	imes 2 	imes 1 = 24$ ways.Within the vowel block,the $4$ vowels can be arranged among themselves in $4!$ ways.$4! = 4 	imes 3 	imes 2 	imes 1 = 24$ ways.Therefore,the total number of arrangements is $4! 	imes 4! = 24 	imes 24 = 576$ ways.

In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?

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