# How to prove this binomial coefficient identity? [duplicate]

I have a dilemma here, how can we show that

Show that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}, 1 \leq r \leq n.$$

I tried solving the right side by substituting everything into the combination's formula but everything gets complicated.. thanks

PS: I tried substituting real valued numbers, and it works, but it should be proof by means of mathematical manipulation.

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 tnx. kannapan sampth – vvavepacket Jan 22 '12 at 13:56

## marked as duplicate by mixedmath♦, Srivatsan, pedja, Davide Giraudo, David MitraJan 22 '12 at 14:01

I think the easiest way to prove this identity is with a combinatorial proof. So we count the same thing in 2 ways. Suppose we have $n$ objects and we are choosing $r$ of them. The left side is that straight out.
Suppose we designate one particular element (it doesn't matter which one, but call it X). Then when choosing $r$ of the $n$ elements, we either have X or we don't. If we do, then we choose $r-1$ of the remaining $n-1$ elements. If we don't, then we choose $r$ of the remaining $n-1$ elements.