How many words can be formed from the letters of the word $\; ARTICLE \;$ so that vowels occupy even places?
The word $\; ARTICLE \;$ has 7 letters with 3 vowels $\; A, \; I, \; E$
There are $3$ even places which the 3 vowels can occupy.
This can be done in $3! = 3 \times 2 \times 1 = 6$ ways
We are left with 4 consonants $\; R, \; T, \; C, \; L$ to occupy the remaining $4$ odd places.
This can be done in $4! = 4 \times 3 \times 2 \times 1 = 24$ ways
$\therefore \;$ Number of words that can be formed from the letters of the word $\; ARTICLE \;$ so that vowels occupy even places $= 6 \times 24 = 144$