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/ |
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