We began by looking at some approaches to the “How many subsets?” problem.

Listing out all of the subsets is a great way for students to get their hands dirty, and it allows for various patterns to emerge from the lists.

Here we have categorized the subsets in a natural way, by **size**. (Note: you can always have fun by arguing whether or not the **empty set** should count!). Lots of great symmetry here in this diagram, and patterns to explore. In particular, this is a great way to illustrate two fundamental principles of **combinatorics**.

First, is the number of ways to **choose r** things from **n**, so it makes sense that this number is the **number of subsets of size r**: to make a subset of size **r** from a set of size **n**, you need to really **choose r **of the **n** available things.

Furthermore, the various lists illustrate the relationship between and . How do you make a subset of size 1? Pick one of the elements from the five. But notice that what’s left over is a subset of size 4! Thus, for every way you can take **one **thing from **five**, there is a corresponding way to take **four **things from **five**.

Another great way to illustrate this is with a group of five students standing in front of the class. Pull one aside: did you make a group of one? Or a group of four?

Lastly, this problem provides a great opportunity to **change your perspective**. Instead of counting the subsets by **size**, think of naming every subset as a word consisting of **Y**s and **N**s. Each letter of the word corresponds to a particular element of the set; **Y **means “Yes, you are in the subset”, and **N** means “No, you are not in the subset”. Here are some examples:

Every subset can be uniquely identified in this manner, and the beauty of this approach is that it is **easy** to count how many five-letter words there are of **Y**s and **N**s: it’s just .

Not only is this a great way to illustrate the **changing your perspective** strategy, but you can also explore the basic idea of a combinatorial argument. In essence, we have counted the same thing (the number of subsets) in two different ways; thus, the ways must be equal! This basic idea gives us one of the fundamental **combinatorial identities:**

In the second half of the session we played around with a **Mandelbrot Team** contest on binary numbers. At the **Mandelbrot Contest** website, you can look at some sample contests (both individual and team), order books of past contests, and register for the official contests.

The individual **Mandelbrot** contests are usually pretty challenging, but they contain great problems, and the team contests are wonderful. The Mandelbrot Team contest takes a rich mathematical problem and breaks it down into 5 or 6** interconnected parts **that range from simple exploration to sophisticated abstraction. These are a great way to build team dynamics in your classes, and a great model for constructing your own guided learning activities, too!

While working through the contest, we had some fun exploring binary representations, binary addition, and binary fractions! Larry showed us a cool way to find a numbers binary representation: the upside-down division method shown here. And he also showed us a fun way to convert fractions into binary by repeated multiplication by 2.