# Thread: Arranging into pairs

1. ## Arranging into pairs

Just need to check if I understand this concept:

In how many different ways can we arrange 2n people into n pairs?

The equivalence classes for this problem would contain $\displaystyle 2^n$ ways to rearrange the pairs and n! ways to order the pairs, correct?

That would leave $\displaystyle \frac{(2n)!}{(2^{n})(n!)}$ as the answer.

I'm having a little trouble grasping the intricacies of partitioning and equivalence classes.

2. Originally Posted by oldguynewstudent
In how many different ways can we arrange 2n people into n pairs?
The equivalence classes for this problem would contain $\displaystyle 2^n$ ways to rearrange the pairs and n! ways to order the pairs, correct?

That would leave $\displaystyle \frac{(2n)!}{(2^{n})(n!)}$ as the answer.

I'm having a little trouble grasping the intricacies of partitioning and equivalence classes.
What exactly is your question?
These are known as unordered partitions.
If we have $\displaystyle m\cdot n$ people to divide into $\displaystyle n$ groups of $\displaystyle m$ each, that can be done in
$\displaystyle \frac{(m\cdot n)!}{(m!)^n(n!)}$ ways.

3. Originally Posted by Plato
What exactly is your question?
These are known as unordered partitions.
If we have $\displaystyle m\cdot n$ people to divide into $\displaystyle n$ groups of $\displaystyle m$ each, that can be done in
$\displaystyle \frac{(m\cdot n)!}{(m!)^n(n!)}$ ways.
Thank you, I just wanted to make sure I understood what I was doing.

This question is part of a section on partitions $\displaystyle \equiv$ equivalence classes.

There was no general formula given.

4. Originally Posted by oldguynewstudent
This question is part of a section on partitions $\displaystyle \equiv$ equivalence classes.
Just one comment: I don’t think this has anything to do with equivalent classes in general.
How do you understand it?

5. Originally Posted by Plato
Just one comment: I don’t think this has anything to do with equivalent classes in general.
How do you understand it?
It is easier to show you the context from pages 37 and 38. This is question 42 on page 38.