Mathematics research question

Anyone have any suggestions on topics i could research for a class I'm taking in the spring?

I have to come up with a topic, research it somewhat extensively, write a paper with proofs of all relivent theorems, and then give a 50 minute presentation of my research to the math department. And I have no clue what I'm going to research...it has to be something relevent to undergraduate mathematics. One of my friends did her's on Cantor's diagonal process and then used it prove that the set of numbers in the golden ratio are countable.

I was thinking about doing something with pascal's triangle but i'm not really sure where I can go with that in terms of proof.

Pascals Triangle has lots of associated proofs that could go along as a research project, however I question the relevance to undergraduate math. Pascal is more of a high school topic IMO. There are so many things you could look at, as a geneticist my mind leaps to statistics applications (eg. Mann-Whitney evolution of Wilcox rank-sum test, and how this has influenced biomedicine etc.). For you, I would recommend Entscheidungsproblem (Hilbert, 1928) as it works with algorithms etc.- much more relevant for an undergraduate audience. There is good controversy to present as well (which makes a 50 min presentation fly by). Alternatively but connected, Gödel's incompleteness theorems are another topic worthy of such a project. Good luck!