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So I was assigned this homework problem: $$\ {s \choose s} + {s+1 \choose s} +...+ {n \choose s} = {n+1 \choose s+1}$$ for all s and all $n \geq s$ I've tried to email both my professor and my TA and their explanations seem contradictory. My professor responded saying the statement I need to prove is "The formula is correct for $0 \leq s \leq n$." Whereas my TA told me I need to use induction on both variables and I'm not sure how to do that. Any help is appreciated!