Discussion

In how many ways can the letters of the word INDIA be arranged, such that all vowels are never together?

Answer: B

INDIA has five letters with I twice, and its vowels are I, I, A.

\[ \text{total} = \frac{5!}{2!} = 60 \]

With the three vowels as one block there are three units (block, N, D):

\[ 3! \times \frac{3!}{2!} = 6 \times 3 = 18 \text{ with the vowels together} \]

\[ 60 - 18 = 42 \]

The repeated I forces a division by \( 2! \) in both counts.

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