Question

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

Answer 1

We have to arrange 7 letters in a row such that vowels occupy even places. There are 3 even places (2, 4, 6). Three vowels can be arranged in these 3 even places in 3! ways. Remaining 4 odd places (1, 3, 5, 7) are to be occupied by the 4 consonants. This can be done in 4! ways. Hence, the total number of words in which vowels occupy even places = $3!\times 4!3\times 2\times 4\times 3\times 2=144$