In the first part of this session, we gave a brief introduction and reminder about binomial coefficients. These help answer questions like How many ways are there to choose a team of 9 starters from a roster of 25 players? The answer can be found by calculating a “falling factorial” that is, multiplying all the numbers from 10 through 25 together, and then dividing by 9!. The full set of binomial coefficients appears in the familiar diagram known as Pascal’s triangle.
In the second part of the session, we investigated how many ways there are to cover an NxN triangular grid using just N rectangles. It turns out the answer can also be expressed using falling factorials and factorials. The sequence of numbers that arises is called the Catalan numbers.