How many even integers lying between 1 to 10000 having distinct digit??
How many $4$ digit even numbers have all $4$ digits distinct ?   This question has been asked if we replace even with positive.   I was solving this Problem like: For first position $1-9$ (No $0$) for next $0-9$ (leaving out the digit which has been choose) for next $0-9$ (leaving out $2$ digits), but on last digit I get stuck. From what I should choose. Will it be $0, 2, 4, 6, 8$, because number needs to be even? How should I solve it?
How to make a div 100% height of the browser window Proving $\sin(x^2)$ Continuous but Not Uniformly Continuous with non-standard Definition Show that the bounded function $f(x) = \sin(x^2)$ is continuous, but not uniformly continuous, on the interval $[0,\infty)$.  The problem should be solved using the following definitions: Definition of Continuous Function Let $X$ and $Y$ be metric spaces. We sat that a function $f:X\rightarrow Y$ is continuous if given any open set $V \subseteq Y$, its inverse image $f^{-1}(V)$ is an open subset of $X$. Definition of Uniformly Continuous Function Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces. If $E \subseteq X$, then we say that a function $f: X\rightarrow Y$ is uniformly continuous on $E$ if for every $\epsilon &gt; 0$ there exists a $\delta &gt; 0$ such that for all $x,y \in E$ we have $$d_x(x,y) &lt; \delta \longrightarrow d_y(f(x),f(y)) &lt; \epsilon$$ For the continuity part it is clear that $f^{-1}(a,b), (a,b) \subseteq (-1,1) = \{x:f(x) \in (a,b)\}$ which appears to be an open set however I doubt this is sufficient as a proof since it holds for any function $f$. Not sure how to go about proving $f$ not uniformly continuous. Seems to be difficult to prove from the definition alone. Grepping through .gz log files Does there exist a magical shell piping which would allow easily to grep through bunch of .gz log files without needing to extract them somewhere?  .gz files are Apache logs, result of log rotation. I'd like to quickly check how often certain URIs are accessed in the past. In my bedroom the bottom half of all receptacles are controlled by one switch. Is it possible to have just one receptacle with top and bottom hot all the time?  The top half is hot all the time the bottom half is controlled by one switch.  Keep in mind there are six other receptacles in this room that are the same way controlled by the same switch. When I took the receptacle out the right side tab had been removed.  The top right screw had a black wire, the bottom right had a red wire, the top and bottom left screws had separate white wires.  I changed the receptacle out as suggested, I capped off the red wire and I capped off the white wire that was attached to the bottom left screw. I attached the other white wire to the top left screw and attached the black wire to the top right screw. I now have none of the receptacles working at all. What did I do wrong? How does the Mac choose which connection to use when both Wifi and Ethernet are connected? When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? I have dual boot configured with Windows 7 (32bit) and Ubuntu 12.04(64bit). I installed Ubuntu after windows, now my windows is corrupt due to some virus and I need to reinstall it. Currently when I start my PC, grub loader comes up and I need to select Ubuntu or Windows.   I was wondering how can I install windows again and still have my Ubuntu install? Is there a way to do this?  I do not want to install Ubuntu again and lose my settings and data there.   Thanks, Ketan How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? It seems like I'm doing something wrong when using foreach variables to indicate node numbers. The two foreach loops are anchoring the paths to the east of each node. Why does this happen?    \documentclass{article}  \usepackage{tikz} \usetikzlibrary{backgrounds,fit,patterns,decorations.pathreplacing,calc}  \makeatletter \tikzset{   labelnode/.style={draw=none, fill=none, black},   latentnode/.style={draw, minimum width=5mm, shape=circle, ultra thick, black},   dagconn/.style={arrows=-&gt;, black, thick}, } \makeatother  \begin{document} \begin{tikzpicture}[scale=1]  \pgfmathsetmacro{\colgap}{1.6}  \pgfmathsetmacro{\rowgap}{1.3}   \node[labelnode] (x1) at (\colgap*0,\rowgap*1) {$X_1$};  \node[labelnode] (x2) at (\colgap*1,\rowgap*1) {$X_2$};  \node[labelnode] (x3) at (\colgap*2,\rowgap*1) {$X_3$};   \node[latentnode,blue] (x4) at (\colgap*0,\rowgap*0) {$X_4$};  \node[latentnode,red] (x5) at (\colgap*1,\rowgap*0) {$X_5$};  \node[labelnode] (x6) at (\colgap*2,\rowgap*0) {$X_6$};   \foreach \c in {1,...,2} {    \foreach \r in {1,...,2} {      \pgfmathsetmacro{\st}{(((\r-1)*3)+\c)}      \pgfmathsetmacro{\en}{\st+1}      \draw[dagconn] ([xshift=0.2cm]x\st) to ([xshift=-0.9cm]x\en);    }  }   \foreach \c in {1,...,3} {    \foreach \r in {1,...,1} {      \pgfmathsetmacro{\st}{((\r-1)*3)+\c}      \pgfmathsetmacro{\en}{\st+3}      \draw[dagconn] (x\st) to (x\en);    }  } \end{tikzpicture} \end{document} Proof for maximal ideals in $\mathbb{Z}[x]$ I have been trying to prove the following theorem:      Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where $p$ is prime integer and $f$ is primitive integer polynomial that is irreducible modulo $p$.    Idea: I tried to establish a homomorphism $\phi: \mathbb{Z}[x] \rightarrow \mathbb{F}$. Since $\mathbb{F}$ is a field it has characteristic $p$ and so integer prime p are mapped to 0 in $\mathbb{F}$. Hence $p\in \ker \phi$. Next we consider $\phi': \mathbb{Z}[x] \rightarrow \mathbb{Z_p}[x]$ and pick an arbitrary maximal ideal $M\in \mathbb{Z}[x]$. So, $\phi'(M)$ is maximal as long as $p \in M$ by correspondence. But now I am stuck at this stage and do not know how to proceed. I guess we might have to use primitivity given in problem but dont know how. Can I check iPhone, iPad, Watch battery on Mac? My goal is to have battery status of other Apple device on my Mac (preferably on menubar). I'd like to have a glimpse of the battery status of my iPhone, iPad, Watch, and Airpods.  I'm a user of Bitbar and iStat menu, but I haven't found any widgets can wirelessly check other Apple device's battery status.  What I know is that I can click the Wi-Fi icon on menu bar and it will show my iPhone battery status. To show the percentage, I have to option + click the Wi-Fi icon and hover over my mouse on the iPhone hot spot section. It's pretty clunky, but I guess at least the API is there.  I am also open to third party apps which can let me see battery status across devices. iOS: Your passcode is required to enable Touch ID Occasionally, when unlocking my iPhone or iPad, I get the following message:     Your passcode is required to enable Touch ID   Now this is not after a number of failed attempts or turning the device off. The device was used within the last 12 hours. Two of the devices are less than 2 months old, and one is a year old. Everything is up to date. It appears to be happening more-or-less at random.  Does anybody know why this message might appear? Triangular Numbers Modulo $k$ - Hit All Values? What's the action of banging two beers together called? When beers in cans or bottles are banged together in rejoice of the company you're with, what is the action called? Is a bool read/write atomic in C# Is accessing a bool field atomic in C#? In particular, do I need to put a lock around:  class Foo {    private bool _bar;     //... in some function on any thread (or many threads)    _bar = true;     //... same for a read    if (_bar) { ... } } How to measure GPU usage? So I've got a file with loads of simple objects. This will later be used to cut out these shapes out of plastic. So, using the material optimally is of outmost importance. I was wondering if there is any script or plugin that could automaticly arrange the objects within the artboard so they could fit best. What should I do when my neural network doesn't generalize well? When NASA sends astronauts to space, they have to wear uncomfortable suits. Why aren't the humans in Star Wars spacecraft doing the same thing? What technology do they have on board that remove this need? Using the Limit definition to find the derivative of $e^x$ Brightness is reset to maximum on every restart The brightness of my laptop is reset to max on every restart. I tried the solution provided at  but had no luck.   This command   cat /sys/class/backlight/acpi_video0/max_brightness   Returns   cat: /sys/class/backlight/acpi_video0/max_brightness: No such file or directory   Then I found that I don't have a folder named acpi_video0, but a folder called intel_backlight:    Every time I increase or decrease the brightness using the brightness control keys, the values in brightness and actual_brightness get updated.  Is there any method I could follow to set the brightness to a fixed value on every boot and vary it as and when I need it using the brightness control keys?