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

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gmat_guy666: Senior | Next Rank: 100 Posts; Posts: 48; Joined: Tue Apr 23, 2013 12:51 am; Thanked: 6 times

by gmat_guy666 » Thu Jun 11, 2015 10:05 am

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

OA B

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Source: Beat The GMAT — Problem Solving | Thu Jun 11, 2015 10:05 am

talaangoshtari: Master | Next Rank: 500 Posts; Posts: 154; Joined: Wed May 21, 2014 4:29 amFollowed by: 1 member; Thanked: 8 times

by talaangoshtari » Thu Jun 11, 2015 10:52 am

C(6,3)=6!/3!3!=20 is the number of ways of choosing 3 boys from 6 boys, and after choosing 3 boys, there are 3 boys left that can form a group. Now, we have 2 groups. Since the order does not matter we should divide it by 2, because it is not important that first we pick which group. So we have 20/2=10

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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 » Thu Jun 11, 2015 10:57 am

gmat_guy666 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

OA B

By the official answer, it appears that the two groups are not unique. That is, there's no group 1 and group 2. There are just two groups of boys.

HOWEVER, for my solution let's PRETEND that there IS a group 1 and a group 2.
Now select 3 boys to be in group 1.
Since the order in which we select the boys doesn't matter, we can use combinations.
We can select 3 boys from 6 boys in 6C3 ways (= 20 ways).
By default, the remaining 3 boys are automatically placed in group 2.

Aside: if anyone is interested, we have a free video on calculating combinations (like 6C3) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Of course there is no group 1 and group 2, so our answer of 20 isn't correct, because the above solution counts each grouping twice.
For example, selecting boys A, B, and C for group 1 (leaving D, E, and F in group 2) is the SAME as selecting boys D, E, and F for group 1 (leaving A, B, and C in group 2), because there is no distinction among the two groups.

Since the initial solution counted each grouping TWICE, the correct answer is 20/2 = [spoiler]10 = B[/spoiler]

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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akash.abhinav: Newbie | Next Rank: 10 Posts; Posts: 4; Joined: Tue Apr 21, 2015 11:10 am; Location: Gurgaon, India

by akash.abhinav » Thu Jun 11, 2015 10:58 am

Say the boys are A,B,C,D,E,F. Now we have to form two groups of 3 boys each.

In order to select 3 boys out of 6 boys, you have 6C3 ways, which on calculation comes out to be 20 ways.

In these 20 ways two ways would be as mentioned below.

1.A B C
2.D E F

But in our question as soon as we select A B C, the other group of D E F is also formed so both these combinations constitute one single way only. Hence the answer is to be divided by 2.

Hence according to me the answer is 10.

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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 » Fri Jun 12, 2015 12:54 pm

gmat_guy666 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

Here's a solution that doesn't require us to divide by 2 at the end.

Let's say the 6 boys are A, B, C, D, E and F
We can create two groups of 3 by combining boy A with two other boys (and the remaining 3 boys are automatically placed in the other group).

So, in how many ways can we assign 2 boys to be the same group as boy A?
There are 5 boys remaining, and we must select 2 to join boy A.
Since the order in which we select the boys doesn't matter, we can use combinations.
We can select 2 boys from 5 boys in 5C2 ways (= 10 ways).

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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