Permutation-Combination
Aptitude

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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.

$\text{None of these}$

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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.