1. ## result

calculate result (clueless)

2. ## Re: result

Originally Posted by Petrus
calculate result
Here it is done.

3. ## Re: result

any possible i can get a smal step by step because i havent really understand this method ( pretty new on it )

4. ## Re: result

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.

5. ## Re: result

Originally Posted by Petrus
any possible i can get a smal step by step because i havent really understand this method
Well I doubt it.
But here is an example. Say $\displaystyle p=10$ then $\displaystyle 10\cdot\binom{22}{10}=10\frac{22!}{(10!)(12!)}~.$

You have to do that 22 times, then add it up.

6. ## Re: result

Its just that i cant sit do this all in a test... if this would be on a test how do u think i can answer this problem?
edit: with none calculate ^^

7. ## Re: result

So- you are expected to know the binomial theorem.

8. ## Re: result

One last question when u rewrite it in 1+2(22)+3(22) i actually did not se where 1 comes from but rest i did get it can u explain?

9. ## Re: result

Originally Posted by HallsofIvy
...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.
This is the ticket right here.

10. ## Re: result

Originally Posted by Petrus
calculate
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.

11. ## Re: result

Originally Posted by Plato
...questions should be posted in an more appropriate forum...
I agree completely. Your conscientious suggestion is exactly what would be expected for a pre-calculus student.

I happen to know from corresponding with the OP that he is in a course involving the calculus.