In how many ways can a group of 6 people be divided into three groups

Question

In how many ways can a group of 6 people be divided into three groups, with each group consisting of 2 people?

A. 12
B. 15
C. 60
D. 90
E. 120

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rajdeep212003 · Answer

One way to do this:

Arrange all 6 people first.

= 6!

But now we are overcounting.

Inside a pair:
AB and BA are same 
So divide by 2! for each pair.

There are 3 pairs:

= divide by (2!)3

Also,
the order of groups does not matter.

(AB)(CD)(EF)
and
(CD)(AB)(EF)

are same grouping.

So divide by 3!

Final:

6! /

= 720 / (8 × 6)

= 15

So answer should be Option B.

– Rajdeep

Adit_ · Answer

6C2*4C2*2C2 = 90.

This includes the arrangements of the 3 groups involved.
Thus 90/3! = 15.

Answer:  Option B

BongBideshini · Answer

Imagine lining up all 6 people:
Total ways =  6! 
This counts every possible ordering.

Each group has 2 people, and inside a group
(A, B) is the same as (B, A)
So for each group, divide by  2! 
There are 3 groups=> divide by  (2!)^3

The groups are identical:
 
 (Group1, Group2, Group3)  
 (Group2, Group1, Group3)   
These are the same arrangement.
So divide by  3!

Final Calculation=
6! ÷  
=15

Ans (B)

ExpertsGlobal5 · Answer

B is the correct answer choice.

Video explanation:

https://www.youtube.com/watch?v=O1Yi8UYiCPk

nequedolore · Answer

6c2*4c2*2c2/3!.

3! since one same selection can be duplicated 6 times.

Hope it helps.

Thanks!

Renchinjav · Answer

I calculated as follows 2C6 * 2C4 * 2C2 = 90 
How can I be sure that 6 should be divided ?

Bunuel · Answer

Because the three groups are  not labeled . For example, the same division AB, CD, EF is counted again as

AB, EF, CD
 CD, AB, EF
 CD, EF, AB
 EF, AB, CD
 EF, CD, AB

So every division is counted 3! = 6 times. Therefore, we divide 90 by 6.

In how many ways can a group of 6 people be divided into three groups

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GMAT Problem Solving (PS) Questions

In how many ways can a group of 6 people be divided into three groups

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards