# Moderate Permutation-Combination Solved QuestionAptitude Discussion

 Q. In how many ways can the letters of the word $EDUCATION$ be rearranged so that the relative position of the vowels and consonants remain the same as in the word $EDUCATION$?
 ✖ A. $\dfrac{9!}{4}$ ✖ B. $\dfrac{9!}{4! × 5!}$ ✔ C. $4! × 5!$ ✖ D. $\text{None of these}$

Solution:
Option(C) is correct

The word $EDUCATION$ is a $9$ letter word, with none of the letters repeating.

The vowels occupy $3^{rd}, 5^{th}, 7^{th}$ and $8^{th}$ position in the word and the remaining $5$ positions are occupied by consonants

As the relative position of the vowels and consonants in any arrangement should remain the same as in the word $EDUCATION$, the vowels can occupy only the afore mentioned $4$ places and the consonants can occupy $1^{st}, 2^{nd}, 4^{th}, 6^{th}$ and $9^{th}$ positions.

The $4$ vowels can be arranged in the $3^{rd}, 5^{th}, 7^{th}$ and $8^{th}$ position in $4!$ Ways.

Similarly, the $5$ consonants can be arranged in $1^{st}, 2^{nd}, 4^{th}, 6^{th}$ and $9^{th}$ position in $5!$ Ways.

Hence, the total number of ways $= \textbf{4! × 5!}$

## (2) Comment(s)

Parul
()

I have a confusion..

There are 5 vowels...

Why have not you included 'E'???

Niks
()

explanation is wrong parul. there are indeed 5 vowels. they can be arranged in 5! ways. and remaining consonants in 4! ways. so total ways is $5! * 4!$.

This makes no difference to final answer, though.