For more inclined mathematicians, i thought of a question for you guys to do. if you want my solution, feel free to either post on my wall or message me and i'll give you my solution.

thought it'd be fun for the more challenging mathematicians on

problem: binomial expansions

prove:

Here's one solution. Recalling the binomial theorem, it's known that



Squaring both sides, we have



The coefficient of the term on the left side is given exactly by , so it suffices to show the coefficient of the same term on the right side is the same. There are 9 ways of obtaining a term containing :







In general, it's the case that



so we have, for instance,



and so adding up all the coefficients from the 9 possibilities, we show that



as required.