@GeorgeSimpson: I disagree with this duplication mark. This question is explicitly asking for a combinatorial proof contrary to the referred question. So, two of three answers of the referred question are to exlude as proper answer for this question. The combinatorial answers here which are quite ok for this question do hardly answer the referred binomial identity. — Markus Scheuer 2 days ago
07:59
@MarkusScheuer Looking among linked and related questions it seems that there are several other questions about combinatorial proof of the same identity. For example: 321022, 497413 1332282. If needed, we can discuss this further in chat. — Martin Sleziak 26 secs ago
Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$ is currently closed as a duplicate of Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$.
As Markus Scheuer explains, the post specifically asks about combinatorial proof, so it probably should not be closed as a duplicate of a question without this restriction.
There are a few more posts about same (or almost the same) identity which specifically ask for combinatorial proofs.
Give the combinatorial proof of the identity $\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$ is not closed at the moment.
Fermat's Combinatorial Identity: How to prove combinatorially? and Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
So I want to ask: Should the posts about combinatorial proofs be reopened (since they have incorrect duplicate target)?
Probably then we should choose a more appropriate duplicate target - one of the combinatorial questions. So there is also question which of the above post should be left open.
I should probably have mentioned that even though the post which is currently chosen as a duplicate target does not explicitly ask for a combinatorial proof, it has an answer which gives a combinatorial proof.
3 hours later…
11:44
Here are some thoughts:
In order to keep questions better manageable it's perfectly fine to mark questions as duplicates as long as they are *really* duplicates. If the marking of duplicates is not appropriate this will only contribute to increase confusion.
A characteristic for being a duplicate is if each appropriate answer for one question is also an appropriate answer for the other question and vice versa. Otherwise it is not a duplicate.
So with respect to the current question in order to be a duplicate of another question this question has also *explicitly* ask for a combinatorial …
In order to keep questions better manageable it's perfectly fine to mark questions as duplicates as long as they are *really* duplicates. If the marking of duplicates is not appropriate this will only contribute to increase confusion.
A characteristic for being a duplicate is if each appropriate answer for one question is also an appropriate answer for the other question and vice versa. Otherwise it is not a duplicate.
So with respect to the current question in order to be a duplicate of another question this question has also *explicitly* ask for a combinatorial …
@MarkusScheuer I will repost your suggestions so that they are clickable. You suggest two pairs of duplicates:
I would choose this one as a duplicate target since it seems to have the most answers and the best answers.
It is questionable whether 321022 should be also closed as a duplicate. Here the poster explicitly stated what they want to count. (So they do not want any combinatorial proof, they specifically want to count subset and "compositions of integers" - even though I do not know what that is.)
a) Vote to reopen and wait whether other users will cast their votes. Then vote to close. (You can attract votes by mentioning posts in chat or on meta, if nothing is happening. But it is worth waiting at least until the post goes through the review queue.)
b) Ask moderators to sort this out. (In this way you do not need 5 reopen votes and then 5 close votes from regular users.)
But I am not sure whether mods would be willing to do something like this unless it is a clear cut case.
You also asked: What's your opinion how to characterize duplicates? I think this was discussed previously on meta. For example this post (and probably few others): What is the definition of duplicate question?
BTW I have noticed that you have tried to ping one of the users who voted to close in your comment. Pings do not work that way: "Users who have closed or reopened a question without a binding close vote (i.e. without a gold tag badge or a moderator vote) cannot be notified." meta.stackexchange.com/questions/43019/…
12:26
@MartinSleziak: I agree with your suggestions regarding the questions, especially regarding [321022](http://math.stackexchange.com/questions/321022) which *is* different (also according to my characterization). (btw compositions are representations of a number as sum of positive integers where the order matters)
I think that a.) is a good way to proceed. I will follow your proposal.
Thanks for link to the Meta question (maybe I will place some thoughts there) and for your info regarding pings.
I think that a.) is a good way to proceed. I will follow your proposal.
Thanks for link to the Meta question (maybe I will place some thoughts there) and for your info regarding pings.
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