Question

Find the total numbers of words that can be made by writing all letters of the word PARAMETER so that no vowel is between two consonants.

Answer 1

The consonants are $P,R,M,T,R$. The vowels are $A,A,E,E$.

By grouping all the consonants as one symbol. Then the number of distinct words can be made out with this symbol and the $4$ vowels are

$\frac{5!}{2!2!}=\frac{5\times 4\times 3\times 2}{2\times 2}=30$

But within this symbol we can be create new words by rearranging the consonants.

This can be done in $=\frac{5!}{2!}=\frac{5\times 4\times 3\times 2}{2}=60$

Thus the total number of words formed is $=30\times 60=1800$.