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Find the sum of the series when n is equal to 83?

$$\binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{\frac{n - 1}{2}} $$

I have got some idea that the trick to solve this particular problem is by using

$\dfrac{83-1}{2} =41$

But I am not getting how?

Thanks in advance.

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I think stackexchange should have a way to send such suggestions privately. (Doesn't look good when it appears to the entire audience) –  Kirthi Raman Mar 14 '12 at 11:18
I agree with that. –  Tomarinator Mar 14 '12 at 11:21
None taken Kannappan Sampath. The policies can be refined as well when we are all participating. It doesn't hurt to suggest new ideas. –  Kirthi Raman Mar 14 '12 at 12:06
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1 Answer

up vote 9 down vote accepted


  • $\displaystyle \sum_{r=0}^n \binom n r=2^n$

  • $\displaystyle \binom n r=\binom n {n-r}$

  • $n$ is odd.

Cook all of these...

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