A good (though tricky) introductory counting problem

My older son is re-working his way through Art of Problem Solving’s Introduction to Counting and Probability. He came across a problem in the review section for chapter 5 that gave him some trouble. I decided to talk through part of it tonight and included my younger son.

My younger son hasn’t been studying any counting lately, so I was expecting the problem to be pretty challenging for him. His work through the first part of the problem is, I think, a nice example of a kid working through a challenging math problem.

The problem is this: How many different ways are there to put 4 distinguishable balls into 3 distinguishable boxes?

The next problem is what was giving my son some trouble: How many different ways are there to put 4 distinguishable balls into 3 indistinguishable boxes?

My younger son struggles with the problem for a bit on this one and then my older son offers his thoughts. What gives my older son a little trouble is the case in which you put 2 balls into one box and 2 into another.

So, after struggling with 2-2-0 case in the last video, we talk about it in a little more depth here. The tricky part is seeing that two cases that don’t look the same are actually the same. It was harder for the boys to see the over counting than I thought it would be. But, we made it!

So, the indistinguishable counting part was pretty confusing to the boys. I think we need to do a few more problems like this to let this particular counting concept sink in.


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