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

Number of letters in INTERMEDIATE is 12.
Number of vowels (A, E, E, I, I, E) i.e is 6.
Number of consonants (T T . R M N D) i.e is 6.
∴ Total words are $\frac{12!}{3!2!2!}$
Now, number of ways of arranging 6 consonants (2 alike) is
$\begin{array}{rcl}\frac{6!}{2!}& =& 6\times 5\times 4\times 3\\ & =& 360\end{array}$
There are 7 gaps in which 6 vowels can be arranged in ⁷P6 ways but 2 are alike of are kind and 3 of other kind
∴ Number of ways of arranging the vowels is ${}^7{P}_6\times \frac{1}{3!21}$
$\begin{array}{rcl}& =& \frac{7!}{1!}\times \frac{1}{3!2!}\\ & =& \frac{7!}{3\times 2\times 2}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 2}\\ & =& 20\times 21\\ & =& 420\\ & & \end{array}$
Hence, the total number of ways when the two vowels never come together is
360 × 420
= 151200