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)

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