What would be a combinatorial approach to find out $\sum\limits_{r=0}^{n}\binom{n+r}{r}$?
What is the result of $\sum\limits_{0}^{n}\binom{n+r}{r}$
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$\begingroup$
combinatorics
summation
binomial-coefficients
combinations
binomial-theorem
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0It seems likely you intended the index of summation to be $r$, ranging from $0$ to $n$. If you like someone can make this edit for you. – 2017-01-20
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0What have you tired so far? – 2017-01-20
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0Considering the ${n+r}\choose{r}$ as elements of the pascal triangle is quite useful. – 2017-01-20
1 Answers
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We have $\sum\limits_{r=0}^n\binom{n+r}{r}=\sum\limits_{r=0}^n\binom{n+r}{n}$. The second sum is equal to $\binom{2n+1}{n+1}$ by the more general hockey stick identity.
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0there are various combinatorial proofs for the hockey stick identity. – 2017-01-20