How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?
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 $=4 ! \times 4 !=576$