How  many ways are there to arrange the letters $A$, $A$, $B$, $B$ around a circle? My first go with was is to arrange $4$ objects around a circle with gets us $(4-1)!$ but we have to take care of the inner arrangements of $A$, $A$ and $B$, $B$ which leaves us with $$\frac 6 {2! \cdot 2!}  $$  but that's obviously wrong. I have tried other methods too but they fail to generalize. By counting by hand for both the $2A$'s and $B$'s case and the $3A$'s and $3B$'s case we get the answers $2$ and $4$ respectively (hopefully I didn't make any mistakes counting)   what's wrong with my argument? what's the right method for solving this? how can this be generalized?
what is the total number of distinct arrangements of $n$ oranges and $n$ apples around a round table? I have no idea how to go about.
Macro for figure position? Why is the following minimal example not working?  I.e., what do I need to do, in order to define the float placement specifier of a figure within a macro? (I know that I should use \newcommand, but in the "full" example the error is part of an involved "superfigure" command. \newcommand is not working too btw.)  \documentclass{scrreprt} \usepackage{blindtext}  \def\PosNotWorking{h} \def\FramboxNotWorking{5cm} \def\CaptionWorking{Caption is working.}  \begin{document}  \Blindtext[1]  \begin{figure}[\PosNotWorking]   \centering     \framebox[5cm]{My special figure. (Pos: \PosNotWorking)}             \caption{\CaptionWorking} \end{figure}  \Blindtext[1]  \end{document}   If I compile this, the figure is at the top (default behaviour). If I enter \figure[h] instead of \figure[\PosNotWorking], the figure is of course between the two paragraphs. Why can't I use Stack Exchange if I'm under 13 years old, or if I'm under 16 years old and from the EU? I was just reading the , and noticed something:  You must be at least 13 years old to access or use the Network or Services, including without limitation to complete a Stack Overflow account registration. By accessing or using the Services or the Network in any manner, you represent and warrant that you are at least 13 years of age. If you are under 13 years old, you may not, under any circumstances or for any reason, access or use the Services or Network in any manner, and may not provide any personal information to or on the Services or Network (including, for example, a name, address, telephone number or email address). If you are located within the European Union, you must be at least 16 years old to access or use the Network or Services, including without limitation to complete a Stack Overflow Account Registration.  I signed up for Stack Overflow when I was around 11 or 12 and I did provide my email address. Are you saying that my use of Stack Overflow was not legal? I'm 16 now (so I'm in the clear), but that just seems a little harsh. Is that possibly related to the age limit for accepting a binding contract/agreement (although I thought that was 18 in the U.S.)? Is there a way for users under the age of 13, or for users from the European Union who are under the age of 16, to use the site legally? Today, comparing processors is obviously not just looking at the frequency and the number of cores.  When choosing a new PC, how do you compare processors? Is there some resource, like a site or something that does (not necessarily super-precise) processor comparisons?  Thanks. How to make the taskbar blink my application like Messenger does when a new message arrive? Send Gmail calendar invite from alternate email How do I find out which DOM element has the focus? I would like to find out, in JavaScript, which element currently has focus. I've been looking through the DOM and haven't found what I need, yet. Is there a way to do this, and how?  The reason I was looking for this:  I'm trying to make keys like the arrows and enter navigate through a table of input elements. Tab works now, but enter, and arrows do not by default it seems. I've got the key handling part set up but now I need to figure out how to move the focus over in the event handling functions. The search parameter isanswered ignores acceptances Find hypotenuse given acute angle bisectors In a right triangle $ABC$ (right-angled at $B$), $D$ and $E$ are points of $\overline{AB}$ and $\overline{BC}$ respectively such that $\overline{CD}$ and $\overline{AE}$ are the angle bisectors of the acute angles of the triangle. Given that $AE=9$ and $CD=8\sqrt{2}$, find the length of the hypotenuse $\overline{AC}$. Revisiting question templates suggested to have templates for tags. Their example was with the  tag on Stack Overflow.  I'm suggesting a more general approach:  Templates, Mark II  I suggest that each site will have a template. That template will prefill the  textbox when a new question is asked. However, it will be completely optional.  Something like (for Stack Overflow):  ## The Problem ## { Please tell us what the problem is with as many details as possible.   Include language, operating system, and any other relevant information.   Remember to add the relevant tags below, too. } ## Attempted Solutions ## { Please tell us what you've tried so far. Have you tried searching Google?    What queries? } ## Existing Code ## { Do you have any existing code to show? Please keep it short and simple.    We should be able to detect the problem right away. }   Why am I suggesting this?  Well, it seems as though most users will stick with the default. They don't bother to read, and so prefilling the question form for them with information that we know might be relevant, will forceWe'll get to that in a second them to actually stop and read.  Scenarios:   Best case: User arrives to ask a question, actually reads the template, fills the correct information in the placeholders. (Yeah, because that'll happen a lot) Neutral: User simply removes all of the template text and rewrites their own question. Result is exact same as we have currently. Worst case: User will ignore or misunderstand the template, will use it wrong or not at all and we'll get something like  &lt;template text here, still unfilled&gt; &lt;(crappy) question&gt;   This can be solved in a couple different ways. We could detect the template tags (i.e. { }, subject to discussion), and throw an error when a user attempts to post with them still available. We could also act more autonomously and remove placeholders which still exist.    Some more points:   Templates should be per-site, not per-tag. Sure, there are differences between the tags, but all questions are expected to have the same general information, and this isn't meant to be a rule, but a guideline to help write better questions. Site templates should be editable by that site's mods, like close reasons. We could autofocus on the first placeholder, and perhaps implement TAB to jump from one placeholder to the other, similar to mixins in IDEs. This doesn't have to be implemented everywhere. This suggestion makes sense to Stack Overflow or Superuser, but not to Anime &amp; Manga or Gaming All of the details are subject to discussion, and all discussion is welcomed. Can a full version of Ubuntu be Installed on a USB Drive? When and why did Harry Potter fall in love with Ginny Weasley? In the earlier HP books it's pretty clear that Harry and Ginny get on quite well as friends (and that she's rather smitten with him) but I'm struggling to identify when and why Harry started to like Ginny in a romantic fashion..  I feel like it is simply presumed that Harry suddenly likes Ginny Weasley back, but do the books actually describe when and why he started to like her that way?  No speculation, please. Canon answers only. I have a 32-bit machine, can I use programs that are meant for 64-bit processors? I am using a 32-bit machine, but I am having to use a relatively obscure semantic database which only comes in 64-bit versions.  What are my options for running it in my local dev environment?  Currently when I try to run it, I get an error message that looks like this:  cannot execute binary file   and when I check my machine info, I get this:  $ uname -a Linux slabrams-desktop 2.6.32-29-generic #58-Ubuntu SMP Fri Feb 11 19:00:09 UTC 2011 i686 GNU/Linux Are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass? In every derivation of Kepler's Laws that I have seen, we assume that the sun is stationary. However, in other places I have read that celestial bodies move about their barycentre (center of mass). So are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass? Evaluate the integral   $$\int\frac{1}{x^2\left(x^4+1\right)^{3/4}}\,dx$$   My Attempt:  Let $x = \frac{1}{t}$. Then $dx = -\frac{1}{t^2}\,dt$. Then the integral converts to   $$ -\int \frac{t^3}{(1+t^4)^{3/4}}\,dt $$  Now Let $(1+t^4) = u$. Then $t^3\,dt = \frac{1}{4}du$. This changes the integral to  $$ \begin{align} -\frac{1}{4}\int t^{-3/4}\,dt &amp;= -u^{1/4}+\mathcal{C}\\ &amp;= -\left(1+t^4\right)^{1/4}+\mathcal{C} \end{align} $$  So we arrive at the solution  $$\int\frac{1}{x^2\left(x^4+1\right)^{3/4}}\,dx = - \left(\frac{1+x^4}{x^4}\right)^{1/4}+\mathcal{C.}$$  Question: Is there any other method for solving this problem? Get the last item in an array Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$? A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used sequences/convergent subsequences and didn't appeal to my intuition. When I saw the problem, my immediate instinct was that an isometry should be "volume-preserving" as well, so the volume of $f(X)$ should be equal to the volume of $X$, which should mean surjectivity if $X$ is compact. The notion of "volume" I came up with was the minimum number of $\epsilon$-balls needed to cover $X$ for given $\epsilon &gt; 0$. If $f$ were not surjective, then because $f$ is continuous this means there must be a point $y \in X$ and $\delta &gt; 0$ so that the ball $B_\delta(y)$ is disjoint from $f(X)$. I wanted to choose $\epsilon$ in terms of $\delta$ and use the fact that an isometry carries $\epsilon$-balls to $\epsilon$-balls, and show that given a minimum-size cover of $X$ with $\epsilon$ balls, that a cover of $X$ with $\epsilon$-balls could be found with one fewer ball if $f(X) \cap B_\delta(y) = \emptyset$, giving a contradiction. Can someone see a way to make this intuition work? Syntax of Using Math Mode: Empty Lines (Beginner Question) I'm just starting to learn LaTeX. I tried typing \begin{equation*}  (math goes here)  \end{equation*}  and I got a whole bunch of errors. It seems LaTeX interpreted the empty line as the equation: when it saw nothing there, I guess it just ended math mode? It then said my other commands needed $'s so I assume that's what happened. When I typed \begin{equation*} (math goes here) \end{equation*}  everything was fine. So I'm confused. If LaTeX sees an empty line in an environment, does it just interpret this as me changing my mind, and deciding not to use the declared environment? Why do we need to discard one dummy variable? Caps-lock remapping stops working I have caps-lock key remapped to control, which usually works fine, but after some uptime and one/more suspend/awake cycles* this mapping breaks, and caps-lock reverts back to the default behavior -- i.e. turning on the led and capitalizing whatever I type. (Even though the Tweaks UI still shows it as being mapped to Ctrl -- see picture below.)  Questions:  Is this a known issue (with maybe a known fix)? (Could not find anything relevant with a quick search.) Is there a way to debug the problem, preferably from the command line?  *Remark: I'm not sure it is the suspend/awake cycle that causes the problem, but I can't connect it to anything else... Remark2: removing and readding the mapping (or rebooting) solves the problem, but it would be nice to find a more permanent solution. I was recently told (  ):     You cannot delete a class in production directly. You will need delete   the class from your sandbox and then deploy the deletions to your   production org.   When you deploy from sandbox to production, the   missing classes will come up in red and you can choose to deploy these   deletions to Production.   However, I'd like some clarification because finding information on how to do this is like pulling teeth, unfortunately.  Am I supposed to:   Delete the Trigger in Sandbox Make a deployment package in Sandbox Add the Trigger which is going to be listed in RED now Deploy (and that deletes it from Production)?   I am not seeing deleted Triggers in Red in Sandbox.  Are they NOT listed under "Triggers" for deployment, but elsewhere?   Frustrated = true.   Does this ONLY apply to Classes??  UPDATE:  I just tried again, with a CLASS, and it still doesn't show up at all as a Deployable package (red or not) after I delete it in the Sandbox.  UPDATE 2:  I was just able to delete a class, that I know was written via IDE from a third party, in both Sandbox and Production.  So now I'm wondering if it has to do with where it was originally deployed from (if from IDE, IDE can delete it - it from Sandbox, you have to jump through hoops).  :-(