Brenda Castro
One of my favorite math moments is when I was confused on a discrete math problem about multisets. I simply did not understand how to combinatorially prove it. $$\sum_{k=0}^m \left({n \choose k}\right) = \left({n+1 \choose m}\right)$$
Now, if this notation is all new to you, don't freak out! You don't have to understand it. I'll share, though, that the idea just expands upon the ideas of combinations and permutations. That is, this part of math is about how many different ways there are to arrange something, like books on a self, letters in a word, or members on a team. Combinations are denoted by \( n \choose k\) and indicate groups where there are no repeats and order doesn't matter (so red, green is the same as green, red). Multisets are denoted by \( \left({n \choose k}\right) \) and indicate groups where there can be repeats and order doesn't matter.
I needed a question that would be answered by each side. I asked, ”How many ways can we choose up to \(m\) scoops of ice cream from a parlor with \(n\) flavors?" Now, I needed to show how both sides counted the number of ways. For the left-hand side, by definition, the number of possibilities is \( \left({n \choose k}\right) \) for \(k\) scoops; we sum up the various multi-sets, giving \( \sum_{k=0}^m \left({n \choose k}\right) \). In short, I just used a definition.
The right-hand side is where I was stuck. I went to my professor and was embarrassed that after a few explanations, I still did not understand. Then, my professor decided to use his backgammon board. He told me to consider each of the triangles on the board as a flavor. I could select up to \(m\) scoops from the \(n\) flavors. Then he told me to imagine one of the triangles as a ”null” flavor. I had to pick \(m\) scoops from these \(n + 1\) flavors. And then it clicked! I could pick any number of scoops because I could just pick the ”null” flavor when I did not want another scoop! I was so giddy when I finally understood this problem, and it felt so great.
In other words, there are two ways to count the number of ways I could get, say, up to 3 scoops of ice cream from the 31 flavors of Baskin Robbins. I could add up all the ways there are to get single, double, and triple scoop cones. (If I want, I can get chocolate, vanilla, and chocolate. If I get vanilla, chocolate, and chocolate, that's the same cone. But vanilla, vanilla, and chocolate is a new cone.) The alternative way is to count only the number of triple scoop cones and pretend I have 32 flavors. One flavor is just a bucket of air. So if I want only strawberry and vanilla, I'll ask for strawberry, vanilla, and air on my cone. The single and double scoop cones are counted thanks to being able to get one or two scoops of air!
I am saddened when I hear people talk about how math ”just isn’t their thing.” I believe that math is for everyone. The problem today is that math tends to be taught in one way with large groups, but different explanations will work for different people. Also, having the ability to ask questions to another person tends to be more effective than searching the internet or through a textbook. I hope to be such a teacher for others; I want to understand where a person is coming from and help them climb to a new level of understanding in mathematics.