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# How many different 6-letter sequences are there that consist

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Manager
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How many different 6-letter sequences are there that consist [#permalink]

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28 Jun 2010, 10:48
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How many different 6-letter sequences are there that consist of 1A, 2B's and 3C's ?

A. 6
B. 60
C. 120
D. 360
E. 720

I thought the answer is 6!/2!3! = 60

But the answer is not 60.
[Reveal] Spoiler: OA

Last edited by Bunuel on 19 Mar 2013, 09:21, edited 1 time in total.
Renamed the topic and edited the OA.
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28 Jun 2010, 11:02
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testprep2010 wrote:

How many different 6-letter sequences are there that consist of 1A, 2B's and 3C's ?

A 6
B 60
C 120
D 360
E 720

I thought the answer is 6!/2!3! = 60

But the answer is not 60.

# of different permutations of 6 letters ABBCCC is indeed $$\frac{6!}{2!3!}=60$$, so you are right answer must be B (60).

THEORY:
Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.
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28 Jun 2010, 11:05
The answer should be 6!/2!3! = 60
What's the source?
Manager
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28 Jun 2010, 11:15
Bunuel , prabir2001 : Thanks for the solution.
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28 Jun 2010, 15:51
great post, bunuel!

Here's an OG12 question where one can apply this concept: page 179, question 191.
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15 Nov 2010, 07:54
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How many different 6-letter sequences are there that consist of 1 A, 2B's, and 3 C's?
(A) 6
(B) 60
(C) 120
(D) 360
(E) 720
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19 Mar 2013, 09:12
There is another approach as we can arrange from either A to C or C to A. So...

6C1 X 5C2 (A to C) = 60; It is equal to 6C3 X 3C2 (C to A) which has the same result.
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19 Mar 2013, 09:20
curtis0063 wrote:
There is another approach as we can arrange from either A to C or C to A. So...

6C1 X 5C2 (A to C) = 60; It is equal to 6C3 X 3C2 (C to A) which has the same result.

Yup this question can be solved a couple of different ways. The easiest is obviously the permutation formula for repeating elements quoted above by Bunuel. The answer is 60, regardless of what the OA indicates.

This highlights a recurring theme on GMAT problems. You should always double check the math for yourself and not trust the material blindly. Typos happen, mistakes occur. If you understand how to solve the questions, you'll get them right on the GMAT as that material is checked, rechecked and checked again for mistakes.

Thanks!
-Ron
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Re: How many different 6-letter sequences are there that consist [#permalink]

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28 Mar 2014, 15:24
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Re: How many different 6-letter sequences are there that consist [#permalink]

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08 Apr 2015, 05:55
Hello from the GMAT Club BumpBot!

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Re: How many different 6-letter sequences are there that consist [#permalink]

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21 Jul 2016, 06:38
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: How many different 6-letter sequences are there that consist   [#permalink] 21 Jul 2016, 06:38
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