How i can find the sum of the series? $\binom{n}{1} + \binom{n}{2} + \cdots+ \binom{n}{\frac{n - 1}{2}} $

Question

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.

Answer 1

Hint:

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

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

• $n$ is odd.

Cook all of these...