A challenging – but illuminating – counting problem

Yesterday “choosing numbers” came up in our summer counting project. The boys seemed to think of them in a fairly formulaic way yesterday. While they were working through a few simple exercises this morning I was trying to think of a question that involved a bit more intuition.

Don’t know what made me think of it, but this pretty well-known problem came to mind:

How many three digit numbers have their digits in strictly increasing order?

When you see this problem for the first time, the connection to choosing numbers is not even remotely obvious. It was fun to hear the boys thinking about this problem and by the end of the project they were able to see the connection.

We started off by talking through some of their initial ideas about how to approach this problem. Those initial thought mainly involve case by case counting.

 

In the next part I asked them to try to think about the problem in terms of choosing numbers. By the end of this discussion the boys are able to understand that to pick a number with increasing digits we need to pick three digits out of 9.

 

Next we looked at a specific case of choosing 3 digits: 3, 6, and 7. What numbers have these digits? How many of those numbers have increasing digits?

At the end of this video the boys are trying to figure out whether or not order matters when we select the three digits.

 

In the last movie we sort out the counting issues – the 84 ways to select 3 numbers out of 9 actually gives the solution to the original problem! I really loved hearing they boys ideas as this approach to the problem suddenly made sense to them.

 

So, a really nice and instructive counting problem. Problems like this show that why counting can be such a fun subject – there’s sometimes much more going on that you might think!

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