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The question is to find the value of:

$n\choose 1$$x(1-x)^{n-1}$ +2.$n\choose2$$x^2(1-x)^{n-2}$ + 3$n\choose3$$x^3(1-x)^{n-3}$ .......n$n\choose n$$x^n$.

I wrote the general term and tried to sum it as:

S=$(1-x)^n$$\sum_{r=1}^n$$r$$n\choose r$$(\frac{x}{1-x})^r$.

I got stuck here.

What do I do after this?

jimjim
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Gokul
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2 Answers2

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HINT: $r\binom{n}r=n\binom{n-1}{r-1}$; this is easy to see if you expand into factorials, and it also has a straightforward combinatorial proof.

Brian M. Scott
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  • M.Scott I am having something to ask .As I find you online now I thought it the right time.How to connect you – Learnmore Jan 28 '15 at 05:23
  • questions are not based on your above answer.The questions are about an interview.Will you entertain that here – Learnmore Jan 28 '15 at 05:26
  • it's an interview conducted by a leading institute of research in algebra i am confused what to start with and what not .i found you were a professor can u guide me please. – Learnmore Jan 28 '15 at 05:38
  • i am reading mathematics all alone.so your thoughts on this can help me a lot – Learnmore Jan 28 '15 at 05:39
  • ok never mind sir,it's always pleasing to talk to you – Learnmore Jan 28 '15 at 05:48
  • @learnmore: I really am sorry. Good luck with the interview! Oh, I just had a thought: it’s possible that someone in chat might have some suggestions, if you’re not too uncomfortable asking there. (I’m going to make the moderators happy and delete my earlier comments in this thread.) – Brian M. Scott Jan 28 '15 at 05:53
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Note that your sum can also be written as $$ \begin{align} \sum_{r=1}^nr\binom{n}{r}x^r(1-x)^{n-r} &=nx\sum_{r=1}^n\binom{n-1}{r-1}x^{r-1}(1-x)^{n-r}\\ &=nx(x+1-x)^{n-1}\\[9pt] &=nx \end{align} $$ This is the expected value of a binomial distribution.

robjohn
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