How many words can be formed out of the letters of the word, 'ORIENTAL',

There are 8 letters in the word ORIENTAL.

We wish to find the total number of arrangements of these 8 letters so that the vowels occupy only odd positions.

There are 4 vowels and 4 odd positions.

These 4 vowels can be arranged in the 4 positions in 4! ways.

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

By fundamental principle of counting:

Total number of arrangements $=4 ! \times 4 !=576$