# Linked Questions

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### Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
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I'm trying to prove this algebraically: $$\sum\limits_{i=0}^{n}\dbinom{n+i}{i}=\dbinom{2n+1}{n+1}$$ Unfortunately I've been stuck for quite a while. Here's what I've tried so far: Turning $\dbinom{n+... 1answer 608 views ### Combinatorial proof for$\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$[duplicate] I'm trying to figure out a combinatorial proof for: $$\displaystyle \sum_{k \mathop = 0}^n \binom {r+k} k = \binom {r + n + 1} n$$ I've tried the committee counting thing, but that didn't work. 1answer 211 views ### How to simplify$\sum_{i=1}^{k}\binom{n + i - 1}{i}$? [duplicate] How to simplify$\sum_{i=1}^{k}\binom{n + i - 1}{i}$? I tried reducing the sum to$\binom{n}{1}, \binom{n}{2}, \binom{n}{3}$and so on but couldn't get a pattern. 2answers 101 views ### Combinatoric Proof of$\sum_0^n\binom{k-1+i}{k-1} = \binom{n+k}{k}$[duplicate]$\sum_0^n\binom{k-1+i}{k-1} = \binom{n+k}{k}$I think there are two different ways to prove the above identity: one is algebraic and the other one is combinatoric. I have seen there's some ways to ... 1answer 96 views ### Prove$\sum_{k=m}^n {k \choose m} = {n+1 \choose m+1}$[duplicate] I am trying to determine whether$\sum_{k=m}^n {k \choose m} = {n+1 \choose m+1}$, so far I am assuming that this is a false statement, but was wondering if there was a proof indicating this is a true ... 1answer 249 views ### Sum of Number of non-decreasing sequences [duplicate] I know that the number of non-decreasing sequences of length$n$and numbers in the sequence lying in the range$[l,r]\$ is given by $$\binom{n+r-l}{n}$$ What is the formula to find the $$\sum_{n=1}^{... 1answer 90 views ### How can you show that \binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}? [duplicate] How can you show that \binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}? This counts the number of subsets from \{1,2,3,\dots,n\} having size 7. To me, the summation part counts subsets of size 6. ... 1answer 71 views ### Solve this combination with a summation. Edited and maybe Solved [duplicate] I have been quite stuck on this problem and with the help of others, I may have solved it. This is what I have to prove.$$\binom{N}{n+1}=\sum^{N-1}_{k=n} \binom{k}{n}.$$So far I have this$${N ...

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