How many different “three of a kinds” are there?

We came to the end of Chapter 4 of our Introduction to Counting and Probability book today. While the boys were working through some of the review problems I was searching for a nice challenge problem for our project today. A question about counting the number of different “three of a kind” poker hands seemed perfect for today’s project.

Well, almost perfect – we don’t really play cards that much so this one required a longer than usual introduction. One important thing in counting “three of a kinds” is that the other two cards cannot match each other – that’s a full house.

Once we find exactly what we want to count, we start by figuring out how to count the number of ways for us to get a group of three matching cards.

For the next part we have to try to find the number of ways to select the remaining two cards. The boys find the right ideas here, but miss that they’ve over counted by a factor of 2. We’ll go back and find the over counting in the next video, here we spent a little extra time making sure the arithmetic was right:

In the last section we compute the value of the expression from the last video. I tell them that the number isn’t right, and they then search through their solution to find where they over counted. Fortunately they found the place where they accidentally double counted.

I end the video telling them about an old work project in we insured a $100 million prize at a poker tournament – the cards we used in this project were from that tournament 🙂

So, a fun project for kids learning about counting and choosing numbers. It is a fun exercise to go through the same calculation for all of the different poker hands, actually. In fact, working through those calculations was the final project for a former student of mine who – what a surprise – did indeed go on to become a professional poker player 🙂