A natural question is: “When is the binomial coefficient an odd number?” Draw a picture, as pretty as you can make it, tabulating the answer for the range $0 ≤ k ≤ n ≤ 15$.
I'm very lost in how I'm supposed to draw this, would this be related to Paschal's Triangle?
Is this picture pretty enough? Here $n$ goes from $0$ at the top to $64$ at the bottom, blue is even and pink is odd.
Yes, it has everything to do with Pascal's triangle. For $n \ge 1$, the binomial coefficient $n \choose k$ is odd if and only if exactly one of the binomial coefficients immediately above it ($n-1 \choose k-1$ and $n-1 \choose k$, omitting any that is not defined) is odd.