The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions, is

Step-by-step solution

A, I, E (3 vowels) M, C, H, N (4 consonants) There are 4 odd places. So, three vowels can be arranged in four places in 4_ P ₃ ways and 4 consonants can be arranged in remaining four places in 4_ P ₄ ways. Hence total number of ways =4_ P ₃ 4_ P ₄ =576