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

Solution(By Examveda Team)

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:
$$\left( {\mathop {}\limits^1 } \right)\left( {\mathop {}\limits^2 } \right)\left( {\mathop {}\limits^3 } \right)\left( {\mathop {}\limits^4 } \right)\left( {\mathop {}\limits^5 } \right)\left( {\mathop {}\limits^6 } \right)\left( {\mathop {}\limits^7 } \right)$$
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
$$\eqalign{ & = {}^4{P_3} \cr & = \left( {4 \times 3 \times 2} \right) \cr & = 24 \cr} $$
4 consonants at the remaining 4 positions may be arranged in $${}^4{P_4} = 4! = $$   24 ways
Required number of ways = (24 × 24) = 576