In how many different ways can the letters of the word MACHINE be arranged so that the vowels may occupy only… · ABC26GN5322

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: (1)(2)(3)(4)(5)(6)(7) Now, 3 vowels can be placed at any of the three places out of four marked 1, 3, 5, 7. Number of ways of arranging the vowels = 4P3 = (4 × 3 × 2) = 24. 4 consonants at the remaining 4 positions may be arranged in 4P4 = 4 = 24 ways. Required number of ways = (24 × 24) = 576.