To find: number of words

Condition: vowels occupy odd positions.

There are 7 letters in the word MACHINE out of which there are 3 vowels namely A C E.

There are 4 odd places in which 3 vowels are to be arranged which can be done P(4,3).

The rest letters can be arranged in 4! ways

Formula:

Number of permutations of n distinct objects among r different places, where repetition is not allowed, is

P(n,r) = n!/(n-r)!

Therefore, the total number of words is

P(4,3)4!× = ×4!

= ×4! = ×24= 576.

Hence the total number of word in which vowel occupy odd positions only is 576.