How many words can be formed using the letter | vedclass

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Question

(B) The word $ARTICLE$ consists of $7$ distinct letters: $A, R, T, I, C, L, E$.There are $3$ vowels $(A, I, E)$ and $4$ consonants $(R, T, C, L)$.In a

Vedclass · Accepted Answer

$144$

Vedclass · Answer

(B) The word $ARTICLE$ consists of $7$ distinct letters: $A, R, T, I, C, L, E$.There are $3$ vowels $(A, I, E)$ and $4$ consonants $(R, T, C, L)$.In a $7$-letter word,the positions are $1, 2, 3, 4, 5, 6, 7$. The even positions are $2, 4, 6$ (total $3$ positions).The $3$ vowels must occupy these $3$ even positions,which can be done in $^3P_3 = 3! = 6$ ways.The remaining $4$ consonants must occupy the $4$ odd positions $(1, 3, 5, 7)$,which can be done in $^4P_4 = 4! = 24$ ways.Total number of words = $3! 	imes 4! = 6 	imes 24 = 144$.

How many words can be formed using the letters of the word $ARTICLE$ if the vowels always occupy even positions?

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