How many words can be formed using the letter | vedclass

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Question

(B) The word '$ARTICLE$' has $7$ distinct letters: $A, R, T, I, C, L, E$.There are $3$ vowels $(A, I, E)$ and $4$ consonants $(R, T, C, L)$.There are

Vedclass · Accepted Answer

$576$

Vedclass · Answer

(B) The word '$ARTICLE$' has $7$ distinct letters: $A, R, T, I, C, L, E$.There are $3$ vowels $(A, I, E)$ and $4$ consonants $(R, T, C, L)$.There are $7$ positions in total. The odd positions are $1, 3, 5, 7$ (total $4$ positions) and the even positions are $2, 4, 6$ (total $3$ positions).Since the $3$ vowels must occupy odd positions,we choose $3$ out of $4$ odd positions and arrange them in $^4P_3$ ways.$^4P_3 = 4 	imes 3 	imes 2 = 24$.The remaining $4$ consonants can be arranged in the remaining $4$ positions in $^4P_4$ ways.$^4P_4 = 4! = 24$.Total number of words = $^4P_3 	imes ^4P_4 = 24 	imes 24 = 576$.

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

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