In how many different ways can the letters of the word MACHINE be arranged so that the vowels may occupy only the odd positions?

Answer: Option B

There are 7 letters in the given word, out of which there are 3 vowels and 4 consonants.
Let us mark the positions to be filled up as follows:
$$\left(\stackrel{1}{}\right)\left(\stackrel{2}{}\right)\left(\stackrel{3}{}\right)\left(\stackrel{4}{}\right)\left(\stackrel{5}{}\right)\left(\stackrel{6}{}\right)\left(\stackrel{7}{}\right)$$
Now, 3 vowels can placed at any of the three places out of four marked 1, 3, 5, 7
Number of ways of arranging the vowels
$$\begin{array}{rl}& ={}^4{P}_3\\ & =\left(4\times 3\times 2\right)\\ & =24\end{array}$$
4 consonants at the remaining 4 positions may be arranged in $${}^4{P}_4=4!=$$   24 ways
Required number of ways = (24 × 24) = 576