Let $$S_r(n)=\sum_{m=0}^n(-1)^m m^r\binom{n}{m}$$ Evaluate $S_n(n)$  For this question, the first part I did is to prove that for integer r, $0&lt;r&lt;n$, $S_r(n)=0$  I did this by induction, given the binomial coefficient $$(1+z)^n=\sum_{m=0}^nz^m \binom{n}{m}$$ and differentiate it $r$ times. The result can be obtained for $S_1(n)=S_2(n)=\dots=S_k(n)=0 \implies S_{k+1}=0, $ given $0&lt;k&lt;n$  I want to show that $S_{n}(n)$ is a linear combination of all $S_k(n)$ for $0&lt;k&lt;n$, but is seems not the case.  How should I proceed?
I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In honesty I've been working on this paper a while now and I'm a bit tired so I'm probably giving up earlier than I usually would.  I won't write out the full question, only the last part:     Let $$S_r(n) = \sum_{m=0}^n\ (-1)^m m^r {n \choose m}$$ where r is a   non-negative integer . Show that $S_r(n)=0$ for $r&lt;n$. Evaluate   $S_n(n)$.   I have shown that $S_r(n)=0$ for $r&lt;n$ by taking $(1+z)^n= \sum_{m=0}^n\ z^m {n \choose m}$, letting $D_r(f(z))=z\frac d{dz}(z\frac d{dz}...(\frac d{dz}(f(z)))...)$ where $z\frac d{dz}$ is applied $r$ times and applied it to both sides.  The left hand side gives a polynomial, degree n which has factor $(1+z)$ for all $r&lt;n$ and the right hand side gives $\sum_{m=0}^n\ z^m (m)^r {n \choose m}$. Setting $z=-1$ yields the required result.   There is some build up to this, so I'm fairly certain that this was the intended method.  I'm stuck however on the very last part. I have tried finding a form for $D_r((1+z)^n)$, but I'm fairly sure that this isn't the correct approach, as the wording of the question implies that $S_n(n)$ needs to be considered separately.  I'm surprised that I didn't find that this question had already been asked, so apologies if it has been.  Thank you.
How to put legend below the chart? I have the following code that produces a line chart:  \begin{figure}[p]     \centering \begin{tikzpicture}     \begin{axis}[     use units,     cycle list name=exotic, %This defines the color. It produces nicer colors         width=15cm, height=8cm,     % size of the image         grid = major,         grid style={dashed, gray!30},         %xmode=log,log basis x=10,         %ymode=log,log basis y=10,         xmin=0,     % start the diagram at this x-coordinate         ymin=0,     % start the diagram at this y-coordinate         %/pgfplots/xtick={0,5,...,60}, % make steps of length 5         %extra x ticks={23},         %extra y ticks={0.507297},         axis background/.style={fill=white},         ylabel=Time,         xlabel=Number of Datapoints (n),         y unit=s,y unit prefix=m,         tick align=outside]        % import the correct data from a CSV file       \addplot table [col sep=comma,trim cells=true,y=Best Case (Ascending)] {datasets/time_mergesort.txt};           \addplot table [col sep=comma,trim cells=true,y=Average Case (Random)] {datasets/time_mergesort.txt};                     \addplot table [col sep=comma,trim cells=true,y=Worst Case (Descending)] {datasets/time_mergesort.txt};                      \legend{Ascending,Random,Descending}     \end{axis}  \end{tikzpicture}     \caption{Mergesort Run Time for Sorting Random Data}     \label{fig:merge_sort_from_csv} \end{figure}   And at the moment the legend appears on the chart on the upper right corner. I need to move that below the chart, or even to leave that on the chart but to move that on the upper left corner.   I have tried the solution mentioned at , but it did not work for me.  How can I move the axis in a different position? What insect is this? (India) Can anyone tell me what kind of insect (if it is one) is this...!! Or is it the pupa of some insect?  Well I don't think that the whole big thing is the insect itself, it appears to be just a kind of protection or shelter which moves along with the tiny little thing that keeps on popping out and going it so as to move. Location:  India I'm currently working on some logging code that supposed to - among other things - print information about the calling function. This should be relatively easy, standard C++ has a type_info class. This contains the name of the typeid'd class/function/etc. but it's mangled. It's not very useful. I.e. typeid(std::vector&lt;int&gt;).name() returns St6vectorIiSaIiEE.  Is there a way to produce something useful from this? Like std::vector&lt;int&gt; for the above example. If it only works for non-template classes, that's fine too.  The solution should work for gcc, but it would be better if I could port it. It's for logging so it's not so important that it can't be turned off, but it should be helpful for debugging. Site.createPortalUser() returning null in test context Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$ Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural numbers.  I am thinking about binomial formula $$\sum_{k=0}^n\binom nk x^{n-k}y^k=(x+y)^n$$ but I'm not sure how to use it.  I find this problem tantalizing because it looks as if there should be some sort of intuitive way so that is why I share it here. I am looking for an answer like  if possible. Can you find one? I'm having problems understanding the following situation. Suppose two 1-tonne cars are going with the same orientations but opposite senses, each 50 km/h with respect to the road. Then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;1\mathrm t\times(50\mathrm{km}/\mathrm h)^2\\&amp;=&amp;2500\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}$$  Now if we look at it from the point of view of one of the cars, then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(0\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;5000\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}.$$  I know that kinetic energy is supposed to change when I change the frame of reference. But I understand that then there must be some other kind of energy to make up for it so that the energy in the system stays unchanged. But I don't see any other kind of energy here. I only see two total energies of the same system that seem to be different. Could you explain this to me?  Please note that while I don't understand any physics, I do understand college level mathematics, so if necessary please use it. (I doubt anything more than high school maths should be needed here, but I want to say this just in case.) In most of the mechanics textbooks , the  is stated without any justification or an experiment to confirm this law. Also after introducing this law forces are also decomposed without any justification. Please suggest any links or books related to this. Any help will be appreciated. Thanks in advance. Can't install MySQL on Mountain Lion This question led to a new package:      I'm trying to make some good-looking numbers in a circle, using the simple command:  \textcircled{1}   However, the circle is misaligned vertically with the number, and look a bit sloppy:    Any suggestions how can I adjust the vertical alignment so that the numbers look more centered w.r.t to the circles? Or, if that's not an option, what other ways to achieve the same results are possible?  I will use those only for numbers, and in very few places, so manual adjustment per glyph is an option.  I'm using pdfLaTeX with Computer Modern, if that matters. What is the strict aliasing rule? When asking about , people sometimes refer to the strict aliasing rule. What are they talking about? remove files when name does NOT contain some words Test if a string contains any of the strings from an array Can apostrophe 's' be used with 'etc.'? cannot start Tomcat 7 server - java.net.BindException: Address already in use Why do statisticians say a non-significant result means "you can't reject the null" as opposed to accepting the null hypothesis? Traditional statistical tests, like the two sample t-test, focus on trying to eliminate the hypothesis that there is no difference between a function of two independent samples. Then, we choose a confidence level and say that if the difference of means is beyond the 95% level, we can reject the null hypothesis. If not, we "can't reject the null hypothesis". This seems to imply that we can't accept it either. Does it mean we're not sure if the null hypothesis is true?   Now, I want to design a test where my hypothesis is that a function of two samples is the same (which is the opposite of traditional statistics tests where the hypothesis is that the two samples are different). So, my null hypothesis becomes that the two samples are different. How should I design such a test? Will it be as simple as saying that if the p-value is lesser than 5% we can accept the hypothesis that there is no significant difference? I'm looking for a fantasy book I read about 10 years ago. I don't remember a lot of it, but there are a few things I still know:   The setting was in medieval times. There was no traveling dimensions. I can't recall whether there was magic or not. It wasn't on Earth, but on some invented world. I believe it was a trilogy, with a focus on the son of some nobility. The goal for the characters was to collect a certain number of stones or gems (I believe there were 5 of them), all in certain colours. Every stone has a certain power attached to it. I believe it started out with one or two characters, but the group grew and every character had a certain colour stone at the end. What is important about the gems is that they are unique. It's not part of a bigger magic system, but they are artefacts from somewhere.   There are some minor incidents I can recall vividly:   They build up a fellowship with some stereotypical fantasy builds: an old school fighter, a minstrel, a female archer and probably more. At one point, they mention that if one of the guards in a country falls asleep on duty, he will be kept awake by his garrison/brothers in arms for seven days. At some point the minstrel falls asleep, and the old fighter teaches him how to stay awake. They travel through a forest full of leeches and other insects to reach one of the stones. Also, I seem to recall there was a member of an underground evil organisation among the nobility, who tried to poison someone and were able to do that because they built up an immunity to the poison.   Does anyone has a clue what this series might be?   Suggestions by kind people, but not what I'm looking for, are:   The Seventh Tower Deltora Quest Diadem Westria Sword of Heavens series I am working on refactoring some old code and have found few structs containing zero length arrays (below). Warnings depressed by pragma, of course, but I've failed to create by "new" structures containing such structures (error 2233). Array 'byData' used as pointer, but why not to use pointer instead? or array of length 1? And of course, no comments were added to make me enjoy the process...  Any causes to use such thing? Any advice in refactoring those?  struct someData {    int nData;    BYTE byData[0]; }   NB It's C++, Windows XP, VS 2003 How do I preload a program for faster start? Is it possible to "preload"(actually not preload, but kinda emulate the startup) e.g. nautilus (or firefox) startup at login, so the first time I launch it myself it launches faster? Probably to start it on another desktop and close after some seconds? This is homework. I've tried my best to find the best answer that I could understand and could be accepted by my lecturer.   I've searched my lecture notes, wiki and google to find some good explanation defining solar irradiance. But somehow I still don't understand how I should 'define solar irradiance'. Can someone give me a few good sentences explaining solar irradiance. What are the changes, if any, in a photovoltaic cell (in terms of short-circuit current and open-circuit voltage) if the irradiance is reduced by 50%? Delay in approval leads to missing out on being awarded an "Explainer" badge