In how many different ways can the letters of the word ‘VARIOUS’ be arranged in such a way that the vowels occupy only the odd positions?

There are 7 different letters in the given word, out of which there are 4 vowels and 3 consonants. Let us mark these positions as under [1][2][3][4][5][6][7] Now, 4 vowels can be placed at any of 4 places out of 4 i.e., 1,3,5,7 Number of ways of arranging the vowels = 4P4 = 4! = 24 ways Also, the 3 consonants can be arranged at the remaining 3 positions Number of ways of these arrangements = 3P3 = 3! = 6 ways Therefore, total number of Ways = 24 × 6 = 144 ways.

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