Question

The number of ways in which the letters of the word "PROPORTION" is arranged without changing the relative positions of the vowels and consonants is

• A. $720$
• B. $\frac{4!}{2!}\cdot 6!$
• C. $\frac{4!}{3!}\cdot 6!$
• D. $\frac{4!6!}{2!3!}$

Answer 1

The correct option is A $720$

In the word PROPORTION there are $6$ consonants of which $2$ are P's, $2$ are R's and the rest different and there are $4$ vowels of which $3$ are O's and rest different.
The position originally occupied by vowels must be occupied by vowels and those occupied by consonants by consonants only.
The vowels must be permuted among themselves and similarly the consonants.
Therefore the consonants can be permuted among themselves in $\frac{6!}{2!2!}$ ways and the vowels can be permuted among themselves in $\frac{4!}{3!}$
Since the two operations are independent, the required number of ways $=\frac{6!}{2!2!}\times \frac{4!}{3!}=720$