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## EVENTS

### How to count with Calculus

#### Tuesday, October 4th, 7pm, Ry352

Abstract: Counting is hard, but calculus is easy! For example, it takes some thought to see that the non-similar binary trees of fixed length are counted by the Catalan numbers, but any freshman walking the halls of Eckhart can tell you that the derivative of x^n is nx^(n-1).

In this talk we will tell you how to show the first statement from the second: how to turn hard counting problems into easy calculus ones. Other examples of problems we'll solve are:

1. N students throw their hats up at graduation. What's the chance that nobody gets their own hat back when they land? By multiplying two Taylor series, we'll find an explicit formula that goes to 1⁄e as N→infinity!

2. It turns out that the Fibonacci numbers are given by the explicit formula F_n = (g^n + 1/g^n)/sqrt(5), where g is the golden ratio. We'll show this by partial fraction decomposition.

3. A drunk man starts at the bar, and stumbles back and forth randomly. On a line and a plane, he will surely return to the bar - but in 3-dimensional space and any higher dimension, this fails to be true! What is even more remarkable is that the solution involves essentially no combinatorics, and is solved by estimating how big a certain integral is.

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### Robby Green

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