The number of ways in which the letters of th | vedclass

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Question

(C) The word $MACHINE$ has $7$ letters: $3$ vowels $(A, I, E)$ and $4$ consonants $(M, C, H, N)$.There are $7$ positions in total,numbered $1$ to $7$.

Vedclass · Accepted Answer

$576$

Vedclass · Answer

(C) The word $MACHINE$ has $7$ letters: $3$ vowels $(A, I, E)$ and $4$ consonants $(M, C, H, N)$.There are $7$ positions in total,numbered $1$ to $7$. The odd positions are $1, 3, 5, 7$,which are $4$ in total.We need to arrange $3$ vowels in these $4$ odd positions,which can be done in $^4P_3$ ways.$^4P_3 = \frac{4!}{(4-3)!} = 4 	imes 3 	imes 2 = 24$ ways.The remaining $4$ letters (consonants) can be arranged in the remaining $4$ positions in $^4P_4$ ways.$^4P_4 = 4! = 4 	imes 3 	imes 2 	imes 1 = 24$ ways.Total number of arrangements $= ^4P_3 	imes ^4P_4 = 24 	imes 24 = 576$.

The number of ways in which the letters of the word $MACHINE$ can be arranged such that the vowels may occupy only odd positions is:

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