We discussed relations and functions, with a bit of review from things that should have been somewhat familiar from algebra. We then discussed equivalence relations and gave some examples and some non-examples.

I introduced the notation

\[nCr = \left({n\atop r}\right).\]

We then used this and showed how it appears in various counting problems. Namely, we have the binomial theorem:

\[(x+y)^n=\left({n\atop 0}\right)x^ny^0+\left({n\atop 1}\right)x^{n-1}y^1+\left({n\atop 2}\right)x^{n-2}y^2+\cdots+\left({n\atop {n-1}}\right)x^1y^{n-1}+\left({n\atop n}\right)x^0y^n.\]

We also considered Pascal’s triangle and Fibonacci numbers, which I will cover in more detail on Monday.


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