Represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient. Came across a probability problem that is sort of challenging for a beginner in a sense that I may have not seen or came across a lot of binomial identities. 
What I am looking for is to see if there is any way to represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient.

Yes. Notice the following:
$$k^2=k\cdot (k-1)+k=2\cdot \frac{k(k-1)}{2}+\binom{k}{1}=2\binom{k}{2}+\binom{k}{1}.$$ Doing this we have
$$\sum _{k=1}^n\left (2\binom{k}{2}+\binom{k}{1}\right )=2\binom{n+1}{3}+\binom{n+1}{2},$$ using the Hockey-Stick identity.
Notice that this is part of a greater picture in which one can go from polynomials like $x^k$ to polynomials like $\binom{x}{k}$ as a change of basis.
