How many arrangements of the letters of the word $'BENGALI'$ can be made i) If the vowels are never together ii) If the vowels are to occupy only odd places

The word BENGALI has 7 distinct letters, out of which 3 are vowels and 4 are consonants. For the first part, the total arrangements are 7! = 5040, and treating the 3 vowels as one unit gives 5! * 3! = 720 arrangements together, making the arrangements with vowels never together 5040 - 720 = 4320. For the second part, there are 4 odd positions to fill with 3 vowels, giving 4P3 = 24 ways, and the remaining 4 consonants can be arranged in the other 4 spots in 4! = 24 ways, resulting in 24 * 24 = 576 ways.

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