The number of words that can be formed out of | vedclass

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(C) 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

(C) 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 $3$ places).The $3$ vowels must occupy these $3$ even places. The number of ways to arrange $3$ vowels in $3$ places is $^3P_3 = 3! = 6$.The remaining $4$ consonants must occupy the $4$ odd positions $(1, 3, 5, 7)$. The number of ways to arrange $4$ consonants in $4$ places is $^4P_4 = 4! = 24$.Total number of words = $^3P_3 	imes ^4P_4 = 6 	imes 24 = 144$.

The number of words that can be formed out of the letters of the word $ARTICLE$ so that the vowels occupy even places is

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