Discrete Math - Day 12

Today was one of those classes that you want to catch on film because it went so well. Thankfully I was taping today's class; I can't wait to view the replay.

We looked at card-based problems today. I introduced the problems by briefly asking if students played card games and then if anyone had ever watched poker tournaments on TV. These tournaments will show probabilities of different hands being drawn. The question to the class was how could you go about calculating these probabilities?

To get things started, we looked at a hand of four cards. I related this to the draw of four cards in Texas Hold'em. The question was how many ways could you get two pair when drawing four cards. Students were off and running, making lists and other representations to get a sense of what was going on.

The solutions being developed presented two schools of thought. The first was thinking that there were 13 choices for the first pair. There would be 12 choices left for the second pair and therefore there would be
     13 x 12 = 156
ways to form two pair.

The second way that students were thinking of the problem was the first choice (say aces) had 12 pairings. Moving to kings, there would now be 11 pairings. The next rank would have only 10 pairings. Continuing down this chain you would finally reach 1 last pairing. The total pairing variations was then
     13 + 12 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78
ways to form two pair.

Many groups were ready to stop there. For these groups I asked how they were accounting for the different suits? Of course this would elicit groans, glares, and other reactions. They would then dive back into it to add this additional wrinkle. Some students thought they could just double their value to account for the other suits that weren't being used. I asked these students if this would account for all of the pairings of the suits? Other groups found there were 6 ways to match suits. I then asked these students what about the suit pairings for the other two cards?

Work continued and students were starting to get results. I asked these students to consider how they could verify or confirm that their solution was correct. I also had students visit different groups to compare results. Many groups were getting 78 x 6 x 6 = 2808 for their answer. One group actually listed out the 36 suit arrangements and had 78 x 36 = 2808 for their answer.

I had students present their results. I started by asking students to show representations and discuss what they were thinking. I tell students I am not interested in an answer, we just want to focus on where they were trying to go. One student showed their list and why they thought there were 13 x 12 = 156 two pair matches. The next presenter showed why they thought there were 78 pairings using the sum referenced above. I asked students what connection there was between these two results since both started by looking at the number of rank pairings that could occur. One student mentioned that 78 x 2 = 156. There were no other thoughts forthcoming so I asked them to continue to consider this as we moved on.

Students next presented different ways that they accounted for suit variations and their final results. These groups consistently were showing a result of 2808.

After the presentations someone asked if there were a more direct way of getting to a result than making lists. I like students to make representations because it helps them to see patterns and structure. I told students this but then asked them to consider the problem. There are thirteen different ranks in a deck of cards. To form two pair we need to use two of these ranks. We are therefore choosing two items out of 13 and do not care about the order they are chosen, i.e. we have a combination.

     ₁₃C₂ = 78

I asked students to compare this value to ₁₃P₂ = 156. The discussion centered on the fact that using a permutation treats AAKK as being different from KKAA when from a card-hand perspective they are exactly the same.

To take into account the number of suit pairings, we have to choose two cards from four. This is another combination that yields

     ₄C₂ = 6

This is done for both ranks so the final solution is

     ₁₃C₂ x ₄C₂ x ₄C₂ = 2808

Students seemed to grasp the connections between what they had done and the use of combinations. To check this understanding we tackled a second problem of how many ways a full-house (3 cards of one rank and 2 cards of another rank) could be drawn in five cards.

I was amazed that one student got the answer in just a couple of minutes. I asked him how he could verify or check to see if his answer was correct. Many other groups were proceeding quickly as well. The main issue that came up was whether to use ₁₃P₂ or ₁₃C₂ to determine how many pairings were able to be created in full-house?

Most groups were correctly accounting for the suit variations using ₄C₃ and ₄C₂ to count the number of ways 3-of-a-kind and 2-of-a-kind can be formed.

I had some students present their thinking and we had results of 3744 or 1872. Students were asking which was right. I asked them to consider the two-pair problem we worked on versus the full-house problem. I listed out AA22 and AAA22 and asked what was a fundamental difference between the two problems?

Many students were stuck at this point. I asked what would happen if the roles of the A and 2 were flipped? The results would be 22AA and 222AA. In the first case we still have the same hand. In the second we have a completely different full-house. While we could use ₁₃C₂ for the two-pair problem, we need to use ₁₃P₂ for the full-house problem since reversing roles changes the outcome. Several students were telling their group that was what they were thinking in the first place but had then changed their minds. I told students to have conviction about their reasoning.

Students were mentally spent but feeling energized about the progress they were making. As one student said, she felt good that they were able to answer the first problem before the class was over, as well she should.

I had students write down their thoughts about the problems worked on today and to capture any take-aways that they had.

Visit the class summary for a student's perspective on the class and to view the lesson slides.