How do I calculate $\sum_{k=1}^{33}\binom{33}{k} k$

Question

I started studying about binom's and sums, How do I calculate $$\sum_{k=0}^{33}\binom{33}{k} k$$

Note: I do know that it is $\binom{33}0\cdot0 + \binom{33}1 \cdot 1 + ... + \binom{33}{33} \cdot 33$, but how do I write it briefly?

Answer 1

Thats a binoarithmetic series the key is to find the expression which has this general term so here its $(1+x)^{33}$ after this we do derivatives and plug in appropriate values of $x$ by trial and error method and then get the answer so its $$\frac{d}{dx}(\sum {33\choose n}.x^n)$$ where $n\in (0,33)$

Answer 2

$$\sum_{k=0}^{33}\binom{33}{k}k=\sum_{k=1}^{33}\frac{33!}{\left(k-1\right)!\left(33-k\right)!}=33\sum_{k=0}^{32}\binom{32}{k}=33\times2^{32}$$