Total number of combination or combinations
Does "the same number of people" behave as singular or plural?
The third formula on the  states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$ where $d = \gcd(m,n)$.  How is this claim justified?  Would we have to use the Chinese Remainder Theorem, as they suggest for proving that $\varphi$ is multiplicative? Counting the number of surjections. How many functions from set $\{1,2,3,\ldots,n\}$ to $\{A,B,C\}$ are surjections? $n \geq 3$  Attempt  I was hoping to count the number of surjections by treating $A,B,C$ like bins, and counting the number of ways to fill them with $1,2,3,\ldots,n$ such that no bins are empty. This is equivalent to putting two "separator" bars in between the $n$ numbers or $n-1 \choose 2$.  However, I think I am missing a step here: prior to putting in the separator bars, I should permutate the $n$ elements. So, multiply by $n!$.  The order of elements in the same bins do not matter. So I should be divided by the number of permutations of the elements in each bins. This is where I am stuck.  Additional Info  I made the question up myself. Feel free to modify the question into a more solvable form if it is too difficult/complicated. White stains and stainless steel cookware I know this question has been asked before in some form or another, but I can't seem to find a definitive answer. I've looked on SA, chowhound, martha stewart, about.com, etc., and answers seem to vary. Answers that I've seen vary from using vinegar to washing using special stainless steel soaps (which I'm not very keen on using) to only using soft sponges and hot water.   The one thing I always do is wash my stainless steel with warm soapy water and a soft sponge. If there's burned food stuck on the cookware, I scrape it off with a wood pot scraper (lightly, of course). And I always dry with a soft cloth immediately after washing. Yet I still get stains!  I'd like to get a general consensus on how to prevent those white and rainbow colored stains as well as minimizing corrosion. Category-theoretic limit related to topological limit? Is it possible to use multiple here-docs in bash? Can one use multiple here-docs to provide input to a command in bash?  $ cat &lt;&lt;&lt;foo &lt;&lt;&lt;bar bar $ cat &lt;&lt;EOF1 &lt;&lt;EOF2 &gt; foo &gt; EOF1 &gt; bar &gt; EOF2 bar   Obviously, in both cases, the second here-doc is used as stdin, and replaces the first reference. Is the solution to use echos instead?  $ cat &lt;(echo -n foo) &lt;(echo bar) foobar   Also, for some reason, using a combination didn't work for me. Why would that be?  $ cat &lt;&lt;&lt;foo &lt;(echo bar) bar $ cat &lt;(echo -n foo) &lt;&lt;&lt;bar foo Jquery - change event firing on page load I generated a key and copied it to my target:  ssh-keygen ssh-copy-id username@hostname   I left the the location of the key as default and did not set any passphrase.  For a small while after that, I was able to ssh username@hostname without being prompted for a password, and then it stopped working for some reason. Inequality $\frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$ Let $a,b,c$ be positive reals . Prove that $\displaystyle \frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$  I found this one in a book, no hints mentioned, but marked as very hard. I can't make any progress... Show that any set in a metric space can be written as the intersection of open sets java.net.SocketException: Connection reset accounting for LED resistance Can't specify the 'async' modifier on the 'Main' method of a console app Trace of a commutator is zero - but what about the commutator of $x$ and $p$? Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about the commutator of the position and momentum operators for a quantum particle? On the one hand: $${\rm Tr}([x,p])~=~0,$$ while on the other hand: $$[x,p]~=~i\hbar.$$ How does this work out? Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: Why is the folder size (in properties) different from the total file/folder sizes in the folder? Here's my problem  I'm trying to do a cleanup of my hard drive, and I'm constantly seeing this scenario:  (1) Aggregate folder size: I check the folder size viewing properties right clicking the folder  (2) Individual folder/file sizes: I check the individual file/folder sizes belonging to that folder viewing properties on the individual file/folder sizes, and then adding them up (with hidden and system files shown).   These two figures are always different, sometimes by a large margin.   For instance, my C drive now, by method (1), shows a consumed space of 42.4 GB.   However, by method (2), by inspecting the respective files/folders, the total consumed space adds up only to 33.1 GB. This is a 28% discrepancy (over the smaller file size) and I have no idea where the rest of the space is going.   My hard drive is constantly in danger of being full (it's bootcamp on a mac with 50 GB of partitioned space), and there's not much I've got installed in terms of software, nor do I have large media files like movies, downloads, temp files, application temp files, torrents, etc.   Most of it's just consumed by the files in C:/Windows, but last I checked, I don't know enough to know which files to delete, and I've heard that you really can't do anything about the space boggling winsxs and system32.   Can you guys recommend a starting point to free up disk space?  Thanks!!  Baggio. Open source computer algebra systems for general relativity I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical aspect rather than numerical computations. Can't install aircrack-ng Whenever I attempt to install aircrack-ng in terminal I get the following error:  Package aircrack-ng is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source However the following packages replace it:   iw  E: Package 'aircrack-ng' has no installation candidate"   I'm new to Ubuntu and so have no idea what to do. How do you make sure email you send programmatically is not automatically marked as spam? This is a tricky one and I've always relied on techniques, such as permission-based emails (i.e. only sending to people you have permission to send to) and not using blatantly spamish terminology.  Of late, some of the emails I send out programmatically have started being shuffled into people's spam folder automatically and I'm wondering what I can do about it.  This is despite the fact that these particular emails are not ones that humans would mark as spam, specifically, they are emails that contain license keys that people have paid good money for, so I don't think they're going to consider them spam  I figure this is a big topic in which I am essentially an ignorant simpleton. Sometimes we read books rather quickly and don't give them much (or any) thought, so the action 'reading' does not necessarily imply that we have given enough thoughts to any book we read.   Is there a word that describes reading a book with great care; usually but necessarily slowly over a long period of time. Login SSH with .ppk file on Ubuntu Terminal I have a production server. To login to the server I must use a .ppk file.  How to login with Ubuntu Terminal and .ppk file?  I tried this :  ssh -i location/file.ppk username@server-ip   but it is showing an error message.