Why Is the Number of Combinations Equal?

Date: 06/13/2007 at 21:15:55
From: Tim
Subject: Explaining combinations

Can you explain why nCr = nC(n-r)?

Date: 06/13/2007 at 22:08:43
From: Doctor Wilko
Subject: Re: Explaining why

Hi Tim,

Thanks for writing to 'Ask Dr. Math'!

Here's an insightful way to think about your problem, without getting
into some rigorous proof of why these must be equal.  

If you take a couple of small examples and think through them, I think
you'll see why it's true.

Think of a simple example, say 3C2,

  3C2 = 3.

If I have three objects and select two of them, there is one object
that remains unselected.  By explicitly choosing two items, I've
implicitly chosen one item that I do not want.

Therefore in this example, 3C2 = 3C1 = 3.

To put this example in a more concrete context, let's say I told you
there are three cans of soda in the fridge.  I ask you how many ways
you can choose two of them for us to drink.

The three cans of soda are: 

  Pepsi  Coke  Sprite  

You could choose:

  1. Pepsi  Coke      (Sprite left in fridge)

  2. Pepsi  Sprite    (Coke left in fridge)

  3. Coke   Sprite    (Pepsi left in fridge)

There are three ways to choose two cans.  Each time you explicitly
chose two cans to take out of the fridge, you implicitly chose one can
to leave in the fridge.  For example, when you chose Pepsi and Coke to
take out of the fridge, you were simultaneously choosing to leave
Sprite in the fridge.

Do you see why 3C2 = 3C1?  They are like "dual" problems that are
happening at the same time.

Any example you make up, you'll see this is true.  If there are 10
students and you want to choose three to be on some committee, you are
simultaneously choosing seven students to not be on the committee.  In
this example,  10C3 = 10C7.

Does this help?

- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/