Please look at the following relation.  $$\forall k \in \mathbb{N}, \ \ \ \ \ \sum^k_{n=0}\frac{(k-n)^k(-1)^n}{(k-n)!n!}=1$$  I don't know a proof of this relation.   I tried mathematical induction. But it didn't work.
If $l$ and $n$ are any positive integers, is there a  proof of the identity     $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!\;$$   which uses the Inclusion-Exclusion Principle?  (If necessary, restrict to the case where $l\ge n$.)    This question is closely related to   and also
How to access the shared folder between two computers running ubuntu I have to computers running Ubuntu and I want to share files between those computer. these computers are already connected through a cross cable and sharing internet connection using firestarter. Both computers have samba installed.   In Desktop computer I have a shared folder How do I access it on the other one. I am new to photography, and need a little advice. I currently have a .   To get rolling, where do I go from here? Does the camera matter? Is this camera enough to start with? What happened to Saruman after he was knifed by Grima? Saruman was one of Maiar, like Gandalf. Meaning that he had a mortal body but presumably, when that body was killed, would simply return back to his ("spirit?") being.  Is there any confirmation from Tolkien that this is indeed what happened after he was knifed by Grima? Or was he somehow completely destroyed since he was de-powered by Gandalf? Solving $f(x+y) = f(x)f(y)f(xy)$ How to perform case-insensitive sorting in JavaScript? How to force a table into page width? I have the following table:  \begin{table}[htb] \begin{tikzpicture} \node (table) [inner sep=0pt] { \begin{tabular}{ l | l }   {\bf Symptom} &amp; {\bf Metric} \\ \hline Class that has many accessor methods and accesses a lot of external data &amp; ATFD is more than a few\\ Class that is large and complex &amp; WMC is high\\ Class that has a lot of methods that only operate on a proper subset of the instance variable set &amp; TCC is low\\ \end{tabular} }; \draw [rounded corners=.5em] (table.north west) rectangle (table.south east); \end{tikzpicture} \caption{God class symptoms} \label{tbl:god_class} \end{table}   Now I want to force the width of the table to be the same as the \textwidth, either by linewrapping of table text or by scaling. How can I achieve that? How to know that your machine learning problem is hopeless? Imagine a standard machine-learning scenario:     You are confronted with a large multivariate dataset and you have a   pretty blurry understanding of it. What you need to do is to make   predictions about some variable based on what you have. As usual, you   clean the data, look at descriptive statistics, run some models,   cross-validate them etc., but after several attempts, going back and   forth and trying multiple models nothing seems to work and your   results are miserable. You can spend hours, days, or weeks on such a   problem...   The question is: when to stop? How do you know that your data actually is hopeless and all the fancy models wouldn't do you any more good than predicting the average outcome for all cases or some other trivial solution?  Of course, this is a forecastability issue, but as far as I know, it is hard to assess forecastability for multivariate data before trying something on it. Or am I wrong?   Disclaimer: this question was inspired by this one  that did not attract much attention. It would be nice to have detailed answer to such question for reference. In  Deathly Hallows, when Harry, Ron, and Hermione are at No. 12, Grimmauld Place, Harry summons Kreacher from the kitchens at Hogwarts where the elf has been working on Harry’s orders given a year previously.  When the trio’s heist in the Ministry of Magic ends with Hermione inadvertently bringing Yaxley inside the perimeter of the Fidelius charm that protects Grimmauld Place, the trio leg it and dare not return there for fear that it’s now crawling with Death Eaters.  Before they leave for the Ministry in the morning, Kreacher is talking about making a stew or pot roast or something for dinner. After the trio go on the run and camp out in the tent, Harry thinks of poor Kreacher preparing a dinner they’ll never return to eat, but they dare not summon him lest a Death Eater tag along (like how Dobby ‘tagged along’ with Kreacher when Harry called him to have him follow Draco in Half-Blood Prince).  The next time we see Kreacher is during the Battle of Hogwarts where he leads the house-elves from the kitchen with battle-cries of “For Master! For Harry Potter!” (or something like that), so he’s clearly lost none of his new-found loyalty to Harry.  Harry is still Kreacher’s master throughout the book, and since he does not communicate at all with Kreacher after their escape from the doorstep, he cannot have ordered Kreacher to return to Hogwarts to work there again.  We know that house-elves can go against their masters’ wishes, but this usually entails them punishing themselves rather brutally, and Kreacher does not seem to have been constantly injuring himself for the better part of a year at Hogwarts.  So when and how does Kreacher leave his master’s house without orders and return to Hogwarts? Note: the eventual outcome of this discussion involved dropping this particular off-topic reason, along with other changes. See:  &nbsp; &mdash;Shog     I was using the reason "demonstrate a minimal understanding" myself quite frequently, but the more I review, the more I have doubts about it and I think this reason can either be dropped entirely or at least rephrased, so I started to use "problems with code you've written must describe the specific problem" instead:     Questions asking for code must demonstrate a minimal understanding of the problem being solved.   So the poster has posted some description, maybe even a snippet of code and in his mind that already proves that he has that minimal understanding required. After all he described the problem and what he tried to solve. So telling him that he lacks that understanding is rather rude, and from the posters point of view doesn't even apply. What both version of the reason not emphasize is that a description of a problem is not various versions of     it doesn't work   but that we rather mean really specific details (see below). I can often see in the reopen review queue that the "improved" version just includes a more verbatim description, while what usually would be expected is some specific error message, a trace or a log, or whatever applies.  Then it continues with:     Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist   comparing it to:     problems with code you've written must describe the specific problem — and include valid code to reproduce it — in the question itself. See SSCCE.org for guidance.   Which says essentially the same, only in more detail.   So the "minimum understanding" reason is IMO a more rude version of the reason telling the user that an SSCCE is missing.  The intent of giving this reason is to tell the user that he needs to give exact details to be able to help him, so IMO a checklist would be much better. Something like:     Your question was closed because it may not provide enough details. Things you can do to improve your question to get an answer include:         Format the code in a way that makes it readable and easily understandable.   Provide the smallest working sample of code demonstrating the problem (link to SSCCE)   Include your input and output.   If applicable provide logs (i.e. Android debugging log, stack trace, etc.).   If applicable provide the exact error messages (compiler, etc.).      And something like this should be best included already in the banner that appears when a question is closed instead of being hidden in some pages you often enough have to actively looking for.  It might be helpful to adjust this, based on the tags or close reason, as I realize that I'm more focused on the programming area, and the above list may not apply to other areas of SO.  Update  Another thing I forgot to mention (but wanted to) is this:     Questions asking for code   Is IMO essentially an invitation of the type of questions     I have a problem, can you provide the solution? (with an optional please)   while     Questions concerning problems with code you've written   puts at least some emphasis on the fact that the poster has to put some effort himself in trying to solve the problem by providing the code which he used to attempt the solution. In a ring, I was trying to prove that for all $a$, $a0 = 0$.  But I found that this depended on a lemma, that is, for all $a$ and $b$, $a(-b) = -ab = (-a)b$.  I am wondering how to prove these directly from the definition of a ring.  Many thanks! What's the deal with the phrase "could care less"?  Whilst growing up, I've always known people (parents etc) to use the phrase "couldn't care less", but I've also come across people who use the phrase "could care less" to mean the same thing (that is, "I do not care at all about that").  To me "couldn't care less" seems a lot more logical. If I couldn't care less about something, that implies I do not care about it at all. In contrast to this, to my mind the phrase "could care less" seems to indicate "I do care about this to some degree at least", but that's definitely not the meaning intended.  What's going on here? Is my (logical) analysis correct or in error? no wifi in Ubuntu 15.04 I cannot see WiFi option in my Ubuntu 15.04. What should I do? Revert asap  lspci -nnk | grep 0280 -A2 output:-  03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30) Subsystem: Foxconn International, Inc. Device [105b:e09a]   rfkill list output:-  0: hci0: Bluetooth Soft blocked: no Hard blocked: no 1: acer-wireless: Wireless LAN Soft blocked: no Hard blocked: no      dmesg | grep ath [    9.545199] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [   10.029684] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [   10.029701] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-5.bin failed with error -2 [   10.029704] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-5.bin': -2 [   10.029711] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-4.bin failed with error -2 [   10.029713] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-4.bin': -2 [   10.029719] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-3.bin failed with error -2 [   10.029721] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-3.bin': -2 [   10.029726] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-2.bin failed with error -2 [   10.029728] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-2.bin': -2 [   10.029733] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware.bin failed with error -2 [   10.029735] ath10k_pci 0000:03:00.0: could not fetch firmware (-2) [   10.029737] ath10k_pci 0000:03:00.0: could not fetch firmware files (-2) [   10.029739] ath10k_pci 0000:03:00.0: could not probe fw (-2) Increasing sequence of step functions and diverging Lebesgue Integrals Let $\{s_n\}$ be an increasing sequence of step functions which converges pointwise on an interval $I$ to a limit function $f$.  If $I$ is unbounded and if $f(x) \geq 1$ everywhere on $I$ except on a set of measure zero, prove that the sequence $\{\int_I s_n\}$ diverges.  (Here $\int_I s_n$ refers to the Lebesgue Integral.)  Conceptually, this makes sense.  Clearly, if $f(x) \geq 1$ almost everywhere on an unbounded interval, then the sequence of Lebesgue integrals should diverge.  However, in proving this, I am running into issues with the convergence of $\{s_n\}$ being pointwise.  If it were uniform, I could easily show this.  How do I show that (even for pointwise convergence) I can choose $n$ large enough to make $\int_I s_n$ as big as I want? When did the red shirt careers change to yellow shirts in Star Trek and why? "Missing return statement" within if / for / while Difference between $ and # in Linux environment Notation: Why write the differential first? From reading answers here, I've noticed that some people write integrals as $\int dx \; f(x)$, while other people write them as $\int f(x)\;dx$.  I realize that there is no mathematical difference between the two notation forms, but was wondering why some people choose the first method over the second.  Is there some place in higher maths that it becomes beneficial to write the differential first?  (I, personally, have always used the second method, just because I was taught that way...) How do you get assembler output from C/C++ source in gcc? How does one do this?  If I want to analyze how something is getting compiled, how would I get the emitted assembly code? Can someone give me some example code that creates a surface with a transparent background in pygame? If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example.