Counting card hands in Discrete Math

In my counting unit, I've used determining card hands for poker every year. This year I have modified this investigation and extended its use. To start things off, I asked students to consider the situation where you are dealt two cards. Their task was to determine how many ways you could be dealt two of the same rank, for example, two kings or two tens.

This task allowed students to focus on a few key concepts without straying too far afield with too little definition. I found that students were much better able to wrestle with key issues by thinking through and representing this problem.

From here, I proceeded to what what have normally been my first card problem: how many way can a hand of four cards contain exactly two pair, that is, two cards of one rank and two cards of a different rank? I asked students to consider the two card hand and how getting two additional cards affects things. I wanted to consider what aspects should be similar or the same and what new elements now needed to be addressed.

I was pleased to see many students readily grasp the similar aspects and then focus on how to deal with the differences that arise in this problem. Several students came to a correct solution rather quickly; much quicker than I have seen solutions arise in the past. Even students who were struggling, were struggling with the aspects that matter rather than going down paths that lead to dead-ends.

With this, I moved to the next aspect, counting the number of ways that a full-house could be dealt in a five-card hand. In this case, students need to consider hands which contain three cards of one rank and two cards of a second rank. Typically, students fail to think about which rank is the three cards and which is the two. Today, very few students did not realize that the order mattered and that three kings and two tens was different from three tens and two kings. Again, solutions were coming to fruition much quicker than I have seen in the past.

Typically, I end the card problems with determining how many ways a five-card hand contains four-of-a-kind. Almost all the students quickly determined this result with many commenting on how easy this problem was. I was pleased with this as this problem usually took longer than it did today.

At this point I departed from my usual direction to address the relative rank and importance of different poker hands. I asked the class which hand should be ranked higher, a full-house or a four-of-a-kind? Based on the fact that four-of-a-kind hands have fewer opportunities of occurring, they decided it should rank higher than a full-house, which it does. I told the class we would be looking at various poker hands and their probability of occurring to rank the importance of poker hands.

This brought up a question from the class as to which poker hand is ranked the highest. Someone suggested a five-of-a-kind but quickly realized this hand was not possible. I mentioned that straight flushes are ranked the highest and a royal flush (10, J, Q, K, A of same suit) is the highest of these hands. Several students said they were curious as to how many ways a straight flush a royal flush could occur. We discussed whether to pursue this right now or later. By a narrow margin students decided they wanted a change of pace, so we pursued a different type of counting problem.

I will revisit the poker hand rankings as we move to discrete probability. This gives students an opportunity to appreciate why, say three of a kind beats two-pair or why a flush beats a straight. We can also pursue what would be the highest ranking hand if a joker is introduced into the deck. There are a lot of variations of this idea that can be pursued and is only limited by available time.

Next year, my discrete math class is being expanded to a full-year course. This will afford the time to pursue some of these investigations and allow the students to develop the poker hand rankings as to which hands should defeat which hands. I think this provides a very real-world application that connects accepted norms in the game of poker to the underlying mathematics that drives those norms.