Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$