In how many ways can the letters of word "EDUCATION&quo

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sat Feb 17, 2018 7:58 am

In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

The OA is D.

Why is D? I'm confused here. Experts, may you give me some help? Please.

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Sat Feb 17, 2018 8:42 am

VJesus12 wrote:In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

The OA is D.

Why is D? I'm confused here. Experts, may you give me some help? Please.

Hello Vjesus12.

Let's take a look at the question.

EDUCATION has 5 vowels and 4 consonants. Hence, there are 5! ways of arranged the vowels and there are 4! ways to arrange the consonants.

Now, let's call A to all the vowels and B all the consonants together. Hence, we have two ways to arrange them, that is to say, AB and BA.

Finally, the total ways of arrange the vowels and consonants as asked are 5!*4!*2 = 5!*4!*2!.

Therefore, the correct answer is the option D.

I hope this answer may help you.

Feel free to ask me again if you have a doubt.

Regards.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Feb 17, 2018 10:12 am

VJesus12 wrote:In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

VOWELS: E, U, A, I, O
CONSONANTS: D, C, T, N

Take the task of arranging the letters and break it into stages.

Stage 1: Arrange the 5 vowels
We can arrange n unique objects in n! ways
So, we can arrange the 5 vowels in 5! ways
In other words, we can complete stage 1 in 5! ways

Stage 2: Arrange the 4 consonants
There are 4 unique objects to arrange.
We can complete stage 2 in 4! ways

Stage 3: The two groups of letters (the vowel group and the consonant group)
There are two options:
1) CONSONANTS-VOWELS
1) VOWELS-CONSONANTS
We can complete stage 3 in 2 ways

By the Fundamental Counting Principle (FCP), we can complete the 3 stages (and thus arrange all 9 letters) in (5!)(4!)(2) ways.

Answer: D
--------------------------

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Feb 18, 2018 3:18 am

VJesus12 wrote:In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!.

Number of ways to arrange the 5 vowels E, A, U, I, and O = 5!.
Number of ways to arrange the 4 consonants D, C, T and N = 4!.
To combine the options above, we multiply:
5!4!.
Since VOWELS-CONSONANTS and CONSONANTS-VOWELS both constitute viable arrangements, the result above must be multiplied by 2:
5!4!2.

The correct answer is D.

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Thu Feb 22, 2018 4:58 pm

VJesus12 wrote:In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels, as well as all the Consonants, always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

We need to determine the number of ways to arrange the letters of word "EDUCATION" such that all the vowels as well as all the consonants always appear together. We start by arranging the given word with vowels together with the consonants together.

[EUAIO] [DCTN]

We first see that there are 2 ways to arrange [EUAIO] and [DCTN]: [EUAIO] [DCTN] and [DCTN] [EUAIO].

Next we must determine the number of arrangements for the letters in [EUAIO] and in [DCTN].

[EUAIO] can be arranged in 5! ways.

[DCTN] can be arranged in 4! ways.

Thus, the total number of ways to arrange the letters of word "EDUCATION" such that all the vowels as well as all the consonants always appear together is 5! x 4! x 2, or 5! x 4! x 2!.

Answer: D

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