The number of words that can be formed using | vedclass

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Question

(D) The word $ARTICLE$ contains $7$ letters in total.The vowels in the word are $A, I, E$,which are $3$ in number.The consonants in the word are $R, T

Vedclass · Accepted Answer

$144$

Vedclass · Answer

(D) The word $ARTICLE$ contains $7$ letters in total.The vowels in the word are $A, I, E$,which are $3$ in number.The consonants in the word are $R, T, C, L$,which are $4$ in number.In a $7$-letter word,the positions are $1, 2, 3, 4, 5, 6, 7$. The even positions are $2, 4, 6$,which are $3$ in total.Since the $3$ vowels must occupy the $3$ even places,the number of ways to arrange them is $^3P_3 = 3! = 6$.The remaining $4$ consonants must occupy the remaining $4$ odd positions $(1, 3, 5, 7)$. The number of ways to arrange them is $^4P_4 = 4! = 24$.Therefore,the total number of words that can be formed is $3! 	imes 4! = 6 	imes 24 = 144$.

The number of words that can be formed using the letters of the word $ARTICLE$,such that the vowels occupy even places,is

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