I want a proof of the identity $${n \choose n} + {n+1 \choose n} + {n+2\choose n} +\cdots + {N \choose n} = {N+1 \choose n+1}$$ where $N\geq n$, by algebraic or combinatoric methods.
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The earlier question has a variety of answers; this one is a combinatorial argument. – Brian M. Scott Nov 06 '16 at 21:56
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In case its not clear why this post is a duplicate, note that yours is $\sum\limits_{t=n}^N\binom{t}{n} = 0+\sum\limits_{t=n}^N\binom{t}{n}=\sum\limits_{t=0}^N\binom{t}{n}$ since $\binom{t}{n}=0$ whenever $t<n$ – JMoravitz Nov 06 '16 at 21:58