Which is the easiest way to prove $\sum \limits_{r=1}^n \binom{2n+1}{r} = 4^n-1$?
I encountered this problem while solving this question:
A student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of books he can select is $63$, then what is the value of $n$?
Evidently this boils to solving for $n$ in $\sum \limits_{r=1}^n \binom{2n+1}{r} = 63$, I used Mathematica to derive that close form. I am wondering how could we get that without electronic aid? I would appreciate an algebraic or combinatorial approach.