How many words can be formed by arranging the letters of the word ‘INDIA’,

To find: Number of words that can be formed so that vowels are never together

Number of words such that vowels are never the together $=$ Total number of words Number of words where vowels are together

Total number of words $=\frac{5 !}{2 !}=60$

To find a number of words where vowels are together

Let the vowels I, I, A be represented by a single letter Z

$\Rightarrow$ the new word is NDZ

A number of permutations $=3 !=6$

Z is composed of 3 letters which can be permuted amongst themselves.

Number of permutations of $Z=\frac{3 !}{2 !}=3$

Number of words where vowels are together $=6 \times 3=18$

$\Rightarrow$ Number of words where vowels are not together $=60-18=42$

There are 42 words where vowels are not together