TITLE: why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
QUESTION [12 upvotes]: Can someone explain to me the proof of $${n+1\choose k} = {n\choose k} + {n\choose k-1}$$?

REPLY [5 votes]: The identity is true by vacuity, because that's how my teacher defines a binomial coefficient. [Actually, one needs to append the base conditions to that for a complete definition.]
On a serious note, please define the notation that you use. As I demonstrated just now, an otherwise interesting question could become trivial for such mundane reasons as differing definitions or conventions.