Question

The number of different words that can be made from the letters of the word INTERMEDIATE, such that two vowels never come together, is……...

Answer 1

Given, the word is INTERMEDIATE
It has vowels (I, E, E, I, A, E) and consonants (N, T, R, M, D, T).
Now we have to arrange these letters if no two vowels come together.
Number of ways to arrange six consonants = \dfrac{6!}{2!}\)
Arrangement of six consonants creates seven gaps.
Number of ways to arrange six vowels in these seven gaps ${=}^7{C}_6\times \frac{6!}{2!3!}\text{ways}$
Therefore, total number of words
$=\frac{6!}{2!}{\times }^7{C}_6\times \frac{6!}{2!3!}$
$=360\times 7\times 60$
$=151200$
Hence, required answer is $151200$.