In how many ways can 4 consonants and 3 vowels be arranged in a row

Question

In how many ways can 4 consonants and 3 vowels be arranged in a row so that the 3 vowels are always together?

A. 360
B. 720 
C. 900
D. 1080
E. 1440

CrackverbalGMAT · Answer

The number of arrangements of n this all taken together is n!.

Taking the 3 vowels as 1 unit there are 5 units to be arranged. (4 consonants + i unit of vowels)

External arrangement = 5! = 120

Internal arrangement of the 3 vowels = 3! = 6

Total number of ways = 120 * 6 = 720

Option B

Arun Kumar

stne · Answer

I thought there were two restrictions :
a) All vowels must be together 
b) First and last must be consonants

4*3!*3!*3

4 = since there are 4 consonants so for the first place we have 4 choices 
3= Now for the last place we have only 3 consonants to choose from.
3!= taking the 3 vowels as 1 unit and 2 remaining consonants we have 3! arrangement 
3!= arranging the vowels among themselves internally.

So total arrangement = 4*3!*3!*3
= 432

Dear  IanStewart  Sir can you please check if my working is correct, thank you.

IanStewart · Answer

I could interpret the question in about five different ways, so I really don't know what it's asking. For one thing, it's not clear if we're arranging  any  three vowels and  any  four consonants (in which case you'd have 21 choices for each consonant, for example), or if we're arranging four given consonants, and three given vowels. We also don't know if the vowels and consonants are distinct, which matters a lot -- if our vowels and consonants are A, A, A, B, B, B, B, for example, and the three vowels need to be together, there are only five possible arrangements in total, and only three if the 'word' must begin and end with a consonant.

And of course, as you point out, it's not even clear what restrictions we're meant to observe here. Your work is perfectly correct if you interpret the question as you did - the answer is 432 if we're given three different vowels, four different consonants, and if we need to arrange all of them so the three vowels appear together and a consonant comes first and last.

stne · Answer

Thank you sir, means a lot, it seems moderator has modified the question to make it clear.

Amity007 · Answer

I got the answer in attempting few different approaches.
But i want to know, why are we not considering the selection of vowels and consonants.
Coz ideally first we have to select 4 consonants from 21 and then 3 vowels out of 5.
So it would be 21c4 x 5c3 x 5! x 3!.
can anyone tell me why are we not using this approach.

Bunuel · Answer

Because the problem doesn’t ask you to  choose  letters. You already have 4 consonants and 3 vowels. You only need to  arrange  them, not select them.

In how many ways can 4 consonants and 3 vowels be arranged in a row

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In how many ways can 4 consonants and 3 vowels be arranged in a row

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