If the letters of the word DAUGHTER are to be arranged so that vowels occupy the odd places, then the number of different words is

DAUGHTER has 8 letters with 3 vowels (A, U, E) and 5 consonants (D, G, H, T, R). The 4 odd positions are 1,3,5,7. Arrange 3 vowels in these 4 positions: 4P3 = 4x3x2 = 24 ways. Arrange 5 consonants in remaining 5 positions: 5! = 120 ways. Total = 24 x 120 = 2880. But checking the calculation: 3 vowels in 4 odd places means 4P3 = 24, and 5 consonants in 5 places = 5! = 120, giving 2880 total arrangements. The answer 576 suggests only 3 odd places or different counting.