calculate result (clueless)
Why clueless? Do you not know what $\displaystyle \begin{pmatrix}22 \\ i \end{pmatrix}$ means? It is the "binomial coefficient" $\displaystyle \begin{pmatrix}n \\ i\end{pmatrix}= \frac{n!}{i!(n- i)!}$.
If you do know that, then this can be done by simple arithmetic:
$\displaystyle \begin{pmatrix}22 \\ 1\end{pmatrix}+ 2\begin{pmatrix}22\\ 2\end{pmatrix}+ \cdot\cdot\cdot+ 21\begin{pmatrix}22 \\ 21\end{pmatrix}+ 22\begin{pmatrix}22 \\ 22\end{pmatrix}$
$\displaystyle = 1+ 2(22)+ \cdot\cdot\cdot+ 21(22)+ 22$
Tedious, but just arithmetic. (I will admit that "$\displaystyle \cdot\cdot\cdot$" hides some pretty big numbers!)
But if you are clever, you will remember that "binomial coefficients" are so named because $\displaystyle (x+ 1)^n= \sum_{i=0}^\infty \begin{pmatrix}n \\ i\end{pmatrix}x^i$ so that, if you take x= 1, $\displaystyle (1+ 1)^n= 2^n= \sum_{i=0}^\infty\begin{pmatrix}n \\ i\end{pmatrix}$.
What about that "p" multiplying the binomial coefficient? Think about the derivative of $\displaystyle (x+ 1)^n= \sum_{i=0}^\infty \begin{pmatrix}n \\ i\end{pmatrix}x^i$ with respect to x.
Here is a comment about the thread and where it was posted.
It is posted in Pre-University algebra forum..
That has to be in most elementary forum on the entire board.
Therefore, I try to answer in that sprite.
If Petrus expects a high level answer then questions should be posted in an more appropriate forum.
I do not think the answer $\displaystyle \sum\limits_{k = 1}^N {k\binom{N}{k}} = N \cdot 2^{N - 1} $ is appropriate for this forum.
Even if it is a well-known fact.