In how many different ways can the letters of | vedclass

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(C) The word '$LEADING$' consists of $7$ distinct letters: $L, E, A, D, I, N, G$.The vowels in the word are $E, A, I$ (total $3$ vowels).To ensure the

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$720$

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(C) The word '$LEADING$' consists of $7$ distinct letters: $L, E, A, D, I, N, G$.The vowels in the word are $E, A, I$ (total $3$ vowels).To ensure the vowels always come together,we treat the group $(E, A, I)$ as a single unit or entity.Now,we have the letters ${L, D, N, G}$ and the single unit $(E, A, I)$.This gives us a total of $4 + 1 = 5$ entities to arrange.These $5$ entities can be arranged in $5!$ ways.$5! = 5 	imes 4 	imes 3 	imes 2 	imes 1 = 120$.Within the group $(E, A, I)$,the $3$ vowels can be arranged among themselves in $3!$ ways.$3! = 3 	imes 2 	imes 1 = 6$.Therefore,the total number of arrangements is $5! 	imes 3! = 120 	imes 6 = 720$.

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

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