Prove that  $$C^n_1 + 2\times C^n_2 + 3\times C^n_3 + \dots + n \times C^n_n = n\times 2^{n-1}$$ I tried the formula  $$C^n_1 + C^n_2 + \dots + C^n_n = 2^n$$ Then , mutiply both sides by $$n$$ $$n\times C^n_1 + n \times C^n_2 +\dots +n\times C^n_n = n\times 2^n$$ I tried to show that the left hand side equals  $$2(C^n_1 + 2 \times C^n_2 + \dots + n\times C^n_n )$$ To get the required but i could not get it ?
Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$ ?    I know that $\sum_{k=0}^{n}\binom{n}{k}= 2^{n}$ and so, $\sum_{k=1}^{n}\binom{n}{k}= 2^{n}-1$ but how to deal with $k$ ?
There seems to be more than a few computer science/programming Stack Exchange networks (is that the correct term?). Stack Overflow, being the first, has by far the most users, questions, and answers.  What is the reasoning for creating the others, and are there clear guidelines for which kinds of questions should be posted? I can see a large amount of potential overflow, many cases of people not getting a good answer to their question, because the person who has the answer isn't browsing that particular network at the moment.  I understand that they were probably created for organizational purposes, but wouldn't it almost make more sense to just have them as categories under Stack Overflow, keep them separated but still connected, instead of making people have to create multiple 'account's, one for each network?  I am sure there was a very good reason to break them up, but as someone that is new to SE, it can be somewhat intimidating to decide which one to post in to ensure you get a good answer. For example, if I am a computer science student, my first instinct might to be to post in the computer science network, until I see that it literally has 1% of the users as the Stack Overflow network, which still seems to be for programming/computer science related questions.  So my second instinct would then to be to post it into the Stack&nbsp;Overflow site on the basis that I have a significantly higher chance of my question being seen and getting a good answer. Then I notice that there is also a 'programmers' network, and I don't even begin to know where that fits in. I, and I am assuming most people, will probably just post on Stack&nbsp;Overflow to be safe.  My question is, other than the short little description blurb of each one, is there a clear cut set of guidelines which what each network is intended for, what kinds of questions should go to each one, and is anything being done to encourage people to post in these newer, smaller (more specialized?) networks as opposed to just posting in big daddy Stack Overflow?  Now again, this is for computer science/programming/"why isn't this code doing what I want?" related questions, I am not saying that if I had a question about Linux or WordPress or something I would have the same confusion. How to test for equality of means for proportions? And two-sided or one-sided? I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you. If I set multiple Alarms, can I tell them apart? Story where people inject "books" in the back of their necks? I read this story in middle school or high school and I cannot for the life of me remember the name of it. I think it started off in a small clan of people, there was a young boy who had just come of age and he had some kind of gift that related to the books? Also I think the injecting was illegal, because I vaguely remember the boy encountering an old man who had a bunch of books that he had to keep hidden. There was also a city and he met a girl there that was Special some how. I may just be merging memories but I think at some point the boy and girl went to an underground city?  Does anyone know what I'm talking about? I get these weird characters when I try to print out a vector element! Find all solutions to $x^2\equiv 1\pmod {91},\ 91 = 7\cdot 13$ I split this into $x^2\equiv 1\pmod {7}$ and $x^2\equiv 1\pmod {13}$.  For $x^2\equiv 1\pmod {7}$, i did: $$ (\pm1 )^2\equiv 1\pmod{7}$$ $$(\pm2 )^2\equiv 4\pmod{7}$$ $$(\pm3 )^2\equiv 2\pmod{7}$$ Which shows that the solutions to $x^2\equiv 1\pmod {7}$ are $\pm1$.  For $x^2\equiv 1\pmod {13}$, i did: $$ (\pm1 )^2\equiv 1\pmod{13}$$ $$(\pm2 )^2\equiv 4\pmod{13}$$ $$(\pm3 )^2\equiv 9\pmod{13}$$ $$ (\pm4 )^2\equiv 3\pmod{13}$$ $$(\pm5 )^2\equiv {-1}\pmod{13}$$ $$(\pm6 )^2\equiv 10\pmod{13}$$Which shows that the solutions to $x^2\equiv 1\pmod {13}$ are $\pm1$.  Thus, I concluded that the solutions to $x^2\equiv 1\pmod {91}$ must be $\pm1$. I thought that $\pm1$ were the only solutions, but apparently I am incorrect! How do I go about finding the other solutions to this congruence? We're working on a Log Viewer. The use will have the option to filter by user, severity, etc. In the Sql days I'd add to the query string, but I want to do it with Linq. How can I conditionally add where-clauses? How can I edit two materials simultaneously? Whenever I open a new shading window, whatever I do in one is just mirrored on the other. If I click a node in one window, the same node and material get selected in the other. I can't edit two shaders at once. How can I edit two shaders simultaneously? Unexpected voltage with switch turned off Does SQLDeveloper support executing scripts? If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$ Let , $\Delta = B(0,1)$.  If $f:\Delta \rightarrow \Delta$ is holomorphic, and $f(0)=0$, prove that $|f(z)+f(-z)| \leq 2|z|^2$.  My attempt so far:  $\displaystyle f(z) = \sum_{n=0}^{+\infty} a_n z^n = \sum_{n=0}^{+\infty} a_{2n} z^{2n} + \sum_{n=0}^{+\infty} a_{2n+1} z^{2n+1}$  $\displaystyle f(-z) = \sum_{n=0}^{+\infty} a_n (-z)^n = \sum_{n=0}^{+\infty} a_{2n} z^{2n} - \sum_{n=0}^{+\infty} a_{2n+1} z^{2n+1}$  Then, $\displaystyle f(z)+f(-z) = 2  \sum_{n=0}^{+\infty} a_{2n} (z^2)^n =: 2 g(z^2)$  If $g(\Delta) \subset \Delta$, as $g(0) = 0$, I can invoke Schwarz Lemma and I win. However, I don't know how to prove this. Could anyone help me? Thanks! Is it correct to say "12:00am"? I've read in various places the first minute of a day described as 12:00am.  Now, whilst I personally prefer to use 24h clock notation and therefore don't have this problem as I can simply describe this minute as 00:00, how is one supposed to express this time using am/pm notation?  Is 00:00am common (strange because there is no 00:00pm), or is the least-bad way to say 12:00am (strange IMHO because it seems to imply that the previous hour was 11:00am when it was actually 11:00pm)? Is it acceptable to use "is become" instead of "has become"? In the King James version of the Bible there is a verse like this:     The Lord is my strength, and my fortress, and my song. And He is become my salvation.   Is it still feasible to use "is become" instead of "has become" and what is the semantic difference? Show bounded and convex function on $\mathbb R$ is constant Should 'fuck' be present in review audits (intentionally fake suggested edit reviews)? Order of elements in $Z_n$ I have this question:     Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$.  Show that the order o($\bar{x}$) of $x \in Z_n$ is   $o(\bar{x})= \frac{n}{HCF(x, n)}$   I've been thinking about it for ages but I still don't get why. A hint would be appreciated. Can I recover app data after the phone has been Factory Reset? Is there a rule for pronouncing “th” at the beginning of a word? How to change radius/weight of curve bezier? try scaling, change radius, but its dont work Potential difference across a zero resistance wire