The word INDIA has 3 vowels I, N and I of which I is repeated twice.
The word INDIA has 2 consonants N and D
(1) For the first arrangement the vowels to occur together
The 3 vowels can be arranged within themselves in three ways.
namely, IIA, AII and IAI
The two consonants and three vowels together can be arranged in 3! x 3 ways = 3 x 2 x 3 =18 different arrangements.
(2) For the arrangement that all vowels do not occur together,
= Total number of arrangements - the number of arrangements when all vowels are together.
Total number of arrangements= 5!/2 (since I is repeating)
= 120/2 = 60
Therefore 60 - 18 = 42 different arrangements.