How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places?

To find: number of words formed

Condition: vowels occupy odd places

There are 8 letters in the word ORIENTAL and vowels are 4 which are O, I, E,A respectively.

There is 4 odd places in which 4 vowels are to be arranged.

The rest 4 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 arrangement is

P(4,4) × 4! = × 4! = ×4! = × 24 = 576.

Therefore, total number of words formed are 576.