It is currently 19 Nov 2017, 19:32

### 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

Author Message
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2757

Kudos [?]: 1911 [0], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

27 Sep 2010, 14:15
1
This post was
BOOKMARKED
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
B

[Reveal] Spoiler: Doubt
Why the answer is not 20? If we select 3 boys out of 6 , we can place them in one group and second group will be automatically selected.

_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1911 [0], given: 235

Math Expert
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132730 [3], given: 12335

### Show Tags

27 Sep 2010, 14:30
3
KUDOS
Expert's post
3
This post was
BOOKMARKED
gurpreetsingh 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)

A. 8
B. 10
C. 16
D. 20
E. 24

[Reveal] Spoiler: OA
B

[Reveal] Spoiler: Doubt
Why the answer is not 20? If we select 3 boys out of 6 , we can place them in one group and second group will be automatically selected.

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 discussed at: combinations-problems-95344.html?hilit=dividing%20objects%20order#p734396

Hope it helps.
_________________

Kudos [?]: 132730 [3], given: 12335

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2757

Kudos [?]: 1911 [0], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

27 Sep 2010, 14:52
thanks I got it
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1911 [0], given: 235

Senior Manager
Joined: 06 Aug 2011
Posts: 388

Kudos [?]: 238 [0], given: 82

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

### Show Tags

24 Sep 2012, 11:59
Bunuel..what if question ask.. order does matter..?? then we wud not divide it by 2! ?

can u little elaborate the order does matter and order does not matter??

_________________

Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !

Kudos [?]: 238 [0], given: 82

Math Expert
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132730 [0], given: 12335

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

### Show Tags

25 Sep 2012, 02:04
sanjoo wrote:
Bunuel..what if question ask.. order does matter..?? then we wud not divide it by 2! ?

can u little elaborate the order does matter and order does not matter??

6-people-form-groups-of-2-for-a-practical-work-each-group-95344.html
probability-85993.html?highlight=divide+groups
combination-55369.html#p690842
probability-88685.html#p669025
combination-groups-and-that-stuff-85707.html#p642634
sub-committee-86346.html?highlight=divide+groups

Hope it's clear.
_________________

Kudos [?]: 132730 [0], given: 12335

VP
Status: Final Lap Up!!!
Affiliations: NYK Line
Joined: 21 Sep 2012
Posts: 1078

Kudos [?]: 661 [0], given: 70

Location: India
GMAT 1: 410 Q35 V11
GMAT 2: 530 Q44 V20
GMAT 3: 630 Q45 V31
GPA: 3.84
WE: Engineering (Transportation)
Re: How many ways are there to split a group of 6 boys into two [#permalink]

### Show Tags

17 Apr 2013, 13:58
Hi
I have a doubt in the question.
6 students can be arranged into 2 groups of 3 each. so ie 2!
Now these 3 students in each group can be arranged in 3! ways in each group so it gives 3!*3!
So total ways in which the group can be arranged = 2!*3!*3! = 72

Can someone help...where did i go wrong

Archit

Kudos [?]: 661 [0], given: 70

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7736

Kudos [?]: 17806 [0], given: 235

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

### Show Tags

18 Apr 2013, 00:15
Archit143 wrote:
Hi
I have a doubt in the question.
6 students can be arranged into 2 groups of 3 each. so ie 2!
Now these 3 students in each group can be arranged in 3! ways in each group so it gives 3!*3!
So total ways in which the group can be arranged = 2!*3!*3! = 72

Can someone help...where did i go wrong

Archit

Responding to a pm:

Two groups are not distinct so you don't have 2!. You did not name the groups as GroupA and GroupB.
(A, B, C) and (D, E, F) split is the same as (D, E, F) and (A, B, C) split.

Also, you do not have to arrange the 3 students in 3! ways. You just have to group them, make a team - not make them stand in a line in a particular sequence.

In fact, you can use the opposite method to understand how to get the answer.
Arrange all 6 in a line in 6! ways.
First 3 is the first group and next 3 is the second group. But guess what, the groups are not distinct so divide by 2!.
Also, since the students needn't be arranged, divide by 3! for each group.

You get 6!/(2!*3!*3!) = 10 (that's how you get the formula)

Also, I have discussed grouping here using different methods: http://www.veritasprep.com/blog/2011/11 ... ke-groups/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 17806 [0], given: 235

Re: How many ways are there to split a group of 6 boys into two   [#permalink] 18 Apr 2013, 00:15
Display posts from previous: Sort by

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

Moderator: Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne 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®.