How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

Question:

How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

Solution:

The word ARTICLE consists of 3 vowels, which have to be arranged in 3 even places. This can be done in 3! ways.

Now, the remaining 4 consonants can be arranged in the remaining 4 places in 4! ways

$\therefore$ Total number of words in which the vowels occupy only even places $=3 ! \times 4 !=144$

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