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I am trying to understand proofs of the $1+2+3+4+ \cdots$ series.

I'm puzzled by the 3rd point of this post where it is solved by binomial coefficient

https://math.stackexchange.com/a/2288/777575

He equates ${n+1 \choose 2}= \frac 12 n(n+1)$ such pairs. Which is true, but what I am confused about is how did he arrive at $\frac 12 n(n+1)$ based on the info of the post: is there a way that I am not seeing?

I'm not talking about reaching the formula through other methods not relevant to how it is solved in the post eg square numbers divided by $2$ etc.

Peter Phipps
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Yozansen
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  • Do you wonder why ${n+1 \choose 2}=\frac{n(n+1)}2$ or why $1+2+\ldots+n={n+1 \choose 2}$? – A.Γ. May 29 '20 at 08:44
  • @A.Γ. The former!!! – Yozansen May 29 '20 at 08:45
  • Isn't it just by the definition? $${n\choose k}=\frac{n!}{k!(n-k)!}$$ Ther are lots of cancelations here. – A.Γ. May 29 '20 at 08:46
  • What I am asking is how did he get 1/2(n)(n+1) from (n+1) choose 2, it does not make sense to me – Yozansen May 29 '20 at 08:49
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    To determine the yellow disc, you pick two blue discs. You can choose 2 blue discs in $n+1 \choose 2$ ways – Robert S May 29 '20 at 08:49
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    $${n+1\choose 2}=\frac{(n+1)!}{2!(n+1-2)!}=\frac{(n+1)\cdot n\cdot (n-1)\cdot (n-2)\cdot\ldots\cdot 1}{2\cdot 1\cdot (n-1)\cdot(n-2)\cdot\ldots\cdot 1}=\frac{(n+1)n}{2}.$$ – A.Γ. May 29 '20 at 08:51
  • @A.Γ. Thanks Wow I was confused I had thought n+1 could not be factorialed – Yozansen May 29 '20 at 08:55
  • Yes,it happens to everybody, no problem. Glad it helped. – A.Γ. May 29 '20 at 08:56
  • $\dbinom{n}{2}$ counts the number of $2$-element subsets of a $n$-element set.

    We can count them: Say our set is ${1,\dots,n}$.

    The $2$-subsets are $\underbrace{{1,2},{1,3},\dots,{1,n}}{(n-1)}$ and $\underbrace{{2,3},{2,4},\dots,{2,n}}{(n-2)}$ and $\underbrace{{3,4},{3,5},\dots,{3,n}}_{(n-3)}$ and $\dots$ and finally $\underbrace{{n-1,n}}_1$.

    That is, there are:

    $$\dbinom{n}{2}=(n-1)+(n-2)+(n-3)+\dots+(n-n)=\frac{n(n-1)}{2}$$

    Many such subsets.

     – Vepir May 29 '20 at 09:14

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