Question

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

Answer 1

There are 4 vowels and 4 consonants the word 'ORIENTAL'. We have to arrange 8 letters in a row such that vowels occupy odd places. There are 4 odd places (1, 3, 5, 7). Four vowels can be arranged in These 4 odd places in 4! ways. Remaining 4 even places (2, 4, 6, 8) 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 odd places = $4!\times 4!$ =$4\times 3\times 2\times 1\times 4\times 3\times 2\times 1$ =576.