# Prove formula by combinatorially [closed]

$$\binom{r}{r} + \binom{r+1}{r}+\binom{r+2}{r} + \cdots + \binom{n}{r}=\binom{n+1}{r+1}$$

I knew that I had to prove from RHS and LHS RHS simply like: take $$r+1$$ out of $$n+1$$ elements But How can I write about LHS?

## closed as off-topic by Eevee Trainer, Shailesh, Leucippus, darij grinberg, Parcly TaxelFeb 27 at 13:53

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• use the combinatorics formula: $\frac{n!}{(n-r)! r!}$ – user29418 Feb 21 at 3:21
• @user29418 can you give more details? – Math askers510520 Feb 21 at 3:27
• $\frac{r!}{0!r!} + \frac{(r+1)!}{1!r!} + ...$, let $n=r, r=r$ for the first one; $n=r+1, r=r$ for the second one, etc. At that point it's all algebra. – user29418 Feb 21 at 3:38
• For an algebraic argument, first expand out the binomials and cancel out factorials. The formula for $\binom nr = \frac{n!}{(n-r)!r!}$ is given above. – астон вілла олоф мэллбэрг Feb 21 at 3:50
• – darij grinberg Feb 21 at 4:15

In choosing $$r+1$$ of elements $$1,2,\dots,n+1,$$ suppose the the last element we choose is element $$k.$$ where $$r+1\leq k\leq n+1.$$ Then we must choose $$r$$ of the preceding $$k-1$$ elements. The total number of ways to do this is the sum on the left-hand side.