How many words can be formed with the letters of the words ORDINATE so the vowels occupy odd places?

Explanation

The given word consists of 8 letters, out of which there arr 4 vowels and 4 consonants.
Let us mark out the positions to be filled up, as shown below:
`(""^(1))(""^(2))(""^(3))(""^(4))(""^(5))(""^(6))(""^(7))(""^(8)).`
Since vowels occupy odd places, they may be placed at 1, 2, 3, 5, 7.
Number of ways of arranging 4 vowels at 4 odd places
`= ""^(4)P_(4)=4! =24.`
The remaining 4 letters of the given word are consonants, which can be arranged at 4 even places marked 2, 4, 6, 8.
Number of ways of arranging 4 consonants at 4 even places
`= ""^(4)P_(4)=4! =24.`
Hence, the total number of words in which the vowels occupy odd places `=(24xx24) =576.`