Can someone explain to me the proof of $${n+1\choose k} = {n\choose k} + {n\choose k-1}$$?
why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
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1Actually I don't really need the explanation of the proof, I just need the proof itself – 2011-11-27
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1http://www.proofwiki.org/wiki/Pascal%27s_Rule – 2011-11-27
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1A proof using binomial theorem can be found here: http://math.stackexchange.com/questions/38900/binomial-expansion – 2011-11-27
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2Actually, you should tell us the definition of binomial coefficient you want to use. Since, in one interpretation, this *is* the definition. – 2011-11-27
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0@Glorfindel possibly I am missing something, but after your edit it seems to me there is not easy way to actually find the duplicate target. This strikes me as unfortunate. . – 2017-04-25
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0@quid you're right. I expected the duplicate to turn up in the notice *below* the post, but it doesn't. You may roll it back if you like. – 2017-04-25
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1@Glorfindel thanks for the reply. I asked for reopen-reclose via a flag instead. Then the end-result will be as hoped for. – 2017-04-25
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0Yes, that will work as well. – 2017-04-25
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0Done. Does it look ok now? – 2017-04-25
3 Answers
The identity is true by vacuity, because that's how my teacher defines a binomial coefficient. [Actually, one needs to append the base conditions to that for a complete definition.]
On a serious note, please define the notation that you use. As I demonstrated just now, an otherwise interesting question could become trivial for such mundane reasons as differing definitions or conventions.
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2One man's definition might well be another man's theorem. – 2012-01-16
I'll just be completely unimaginative here:
$$\eqalign{{n\choose k}+{n\choose k-1}&= {n!\over (n-k)!k!}+ {n!\over (n-(k-1))! (k-1)!}\cr &={n!\over (n-k)!k!}+ {n!\over (n-k+1)! (k-1)!}\cr &={(n-k+1)n!\over(n-k+1) (n-k)!k!}+ {n!k\over (n-k+1)! (k-1)! k}\cr &={(n-k+1)n! + n!k\over (n-k+1)! k!}\cr &={n\cdot n !-kn!+n!+n!k\over (n-k+1)! k!}\cr &={n\cdot n ! +n! \over (n-k+1)! k!}\cr &={n!(n+1) \over (n-k+1)! k!}\cr &={(n+1)! \over( (n+1)-k)! k!}\cr &={n+1\choose k}. }$$
One proof goes like this: Suppose you have a list of all $\dbinom {n}{k-1}$ ways to choose $k-1$ objects out of $n$. Then we add an $(n+1)^\text{th}$ object to those from which we can choose. How do we make a list of all $\dbinom{n+1}{k}$ ways to choose $k$ out of these $n+1$? Here's how: first make a list of all ways to choose $k$ out of the original $n$ objects. That's a partial list. All of its items exclude the "new" object. It has $\dbinom nk$ items. Then take the list you already had, of all ways to pick $k-1$ out of those $n$. To each item on the list, giving $k-1$ objects, you add the "new" object, getting a set of $k$ of the $n+1$. That's another partial list. All of its items include the "new" object. It has $\dbinom{n}{k-1}$ items.
Now just add the two partial lists together: the list of those that exclude the "new" object—there are $\dbinom nk$ of those—and the list of those that include the "new" object—there are $\dbinom{n}{k-1}$ of those.