How many ways are there to split a group of 6 boys into two : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 23 Jan 2017, 21:59

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# How many ways are there to split a group of 6 boys into two

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Intern
Joined: 13 Nov 2010
Posts: 23
Followers: 1

Kudos [?]: 9 [0], given: 0

How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

25 Nov 2010, 16:06
9
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

47% (01:28) correct 53% (00:44) wrong based on 200 sessions

### HideShow timer Statistics

How many ways are there to split a group of 6 boys into two groups of 3 boys each? (The order of the groups does not matter)

A. 8
B. 10
C. 16
D. 20
E. 24
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 09:15, edited 1 time in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7102

Kudos [?]: 93592 [0], given: 10579

Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

### Show Tags

25 Nov 2010, 16:13
Expert's post
8
This post was
BOOKMARKED
yufenshi wrote:
How many ways are there to split a group of 6 boys into two groups of 3 boys each? (The order of the groups does not matter)

8
10
16
20
24

The official answer is B, but I don't understand why in this case we need to divide by 2. It seems to me to be the same questions as: how many ways to choose 3 people out of 6 people. in that case it would be 20.

Thanks!

GENERAL RULE:
1. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is important is $$\frac{(mn)!}{(n!)^m}$$

2. The number of ways in which $$mn$$ different items can be divided equally into $$m$$ groups, each containing $$n$$ objects and the order of the groups is NOT important is $$\frac{(mn)!}{(n!)^m*m!}$$.

BACK TO THE ORIGINAL QUESTION:
In original question as the order is NOT important, we should use second formula, $$mn=6$$, $$m=2$$ groups $$n=3$$ objects (people):
$$\frac{(mn)!}{(n!)^m*m!}=\frac{6!}{(3!)^2*2!}=10$$.

This can be done in another way as well: $$\frac{C^3_6*C^3_3}{2!}=10$$, we are dividing by $$2!$$ as there are 2 groups and order doesn't matter.

For example if we choose with $$C^3_6$$ the group {ABC} then the group {DEF} is left and we have two groups {ABC} and {DEF} but then we could choose also {DEF}, so in this case second group would be {ABC}, so we would have the same two groups: {ABC} and {DEF}. So to get rid of such duplications we should divide $$C^3_6*C^3_3$$ by factorial of number of groups - 2!.

This concept is also discussed at:
combinations-problems-95344.html?hilit=dividing%20objects%20order#p734396
split-the-group-101813.html?hilit=split
9-people-and-combinatorics-101722.html?hilit=divided%20equally%20into#p788744
ways-to-divide-99053.html?hilit=divided%20equally%20into#p763471

Hope it helps.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7130
Location: Pune, India
Followers: 2140

Kudos [?]: 13706 [0], given: 222

Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

### Show Tags

25 Nov 2010, 20:49
and after going through Bunuel's explanation, to ensure that you have understood the concept, try the question in this post: http://gmatclub.com/forum/combination-105384.html

It the same reason why you do not multiply by 2 in the question in this post.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 08 Nov 2010 Posts: 417 WE 1: Business Development Followers: 7 Kudos [?]: 106 [0], given: 161 Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink] ### Show Tags 27 Nov 2010, 12:24 if i used the 6!/3!*(6-3)! * 1/2 is that ok as well? _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7130 Location: Pune, India Followers: 2140 Kudos [?]: 13706 [0], given: 222 Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink] ### Show Tags 27 Nov 2010, 12:44 144144 wrote: if i used the 6!/3!*(6-3)! * 1/2 is that ok as well? Absolutely! You are doing 6C3 * (1/2). _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 36618
Followers: 7102

Kudos [?]: 93592 [0], given: 10579

Re: ways to split a group of 6 boys into two groups of 3 boys ea [#permalink]

### Show Tags

27 Nov 2010, 13:12
VeritasPrepKarishma wrote:
144144 wrote:
if i used the
6!/3!*(6-3)! * 1/2 is that ok as well?

Absolutely! You are doing 6C3 * (1/2).

I'd say it depends on the logic behind this formula.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13533
Followers: 577

Kudos [?]: 163 [0], given: 0

Re: How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

17 Aug 2014, 03:17
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).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13533
Followers: 577

Kudos [?]: 163 [0], given: 0

Re: How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

18 Aug 2015, 05:33
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).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Intern
Joined: 29 Aug 2015
Posts: 1
Followers: 0

Kudos [?]: 0 [0], given: 73

Re: How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

26 Nov 2015, 15:40
Or simply put it in an anagram grid as: BBB|BBB , so we have 6!/3!*3!*2!=6*5*4/3*2*2=10.
Re: How many ways are there to split a group of 6 boys into two   [#permalink] 26 Nov 2015, 15:40
Similar topics Replies Last post
Similar
Topics:
In how many different ways can a group of twelve people be split into 1 15 Oct 2016, 13:17
3 In how many ways can be 6 boys and 6 girls sit around circular table 3 22 Jan 2015, 06:52
13 How many ways can you group 3 people from 4 sets of twins if no two 7 05 Oct 2014, 00:00
6 How many ways are there to split a group of 6 boys into two 16 06 Sep 2011, 08:35
17 If a choir consists of 5 boys and 6 girls, in how many ways can the 12 10 Oct 2007, 22:36
Display posts from previous: Sort by

# How many ways are there to split a group of 6 boys into two

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.