How many words can be formed using the letter | vedclass

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(D) The word $MAXIMUM$ contains $7$ letters: $M, A, X, I, M, U, M$.The consonants are $M, X, M, M$ (total $4$) and the vowels are $A, I, U$ (total $3$

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) The word $MAXIMUM$ contains $7$ letters: $M, A, X, I, M, U, M$.The consonants are $M, X, M, M$ (total $4$) and the vowels are $A, I, U$ (total $3$).To ensure no two vowels are together,we use the gap method.First,arrange the consonants: $M, X, M, M$. The number of ways to arrange these is $\frac{4!}{3!} = 4$.There are $5$ possible gaps created by these consonants: $\_ M \_ X \_ M \_ M \_$.We need to place $3$ vowels in these $5$ gaps,which can be done in $P(5, 3) = 5 	imes 4 	imes 3 = 60$ ways.Total arrangements = $4 	imes 60 = 240$.Since $240$ is not equal to $4! = 24$,$3! 	imes 4! = 144$,or $7! = 5040$,the correct answer is $None \ 	ext{of these}$.

How many words can be formed using the letters of the word $MAXIMUM$ such that no two vowels are together?

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