I wanted to know, how can i solve this.  For a given positive integer n, how many solutions $(x,y)$ exist for $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ with $x$ and $y$ being positive integers and $(x \neq y)$.
I have a question about the following problem from a Putnam review:  Let $n\in \mathbb{N}$. Find how many pairs of natural numbers $(x, y)\in \mathbb{N}\times \mathbb{N}$ solve $$ \frac{xy}{x+y}=n. $$  I have found some solutions for all $n$, such as $(2n, 2n)$ and $(n+1, n(n+1))$, but I feel as though this is the wrong approach, as the question is only asking for the number of solutions to the equation.  I don't really want a complete solution, but any hints would be greatly appreciated.
Probability of a 2 pair in poker What is wrong with my method of find the probability of a 2 pair?  For the first card we have 52 choices, one this is picked( call it A) we have 3 choices(different suite than A but this same value) for the second card, ( lets call this card B). Now we have AB, to we have to throw away 2 more cards so we don't have 3 of a kind or a four of a kind. Therefore one our third card we have 48 choices and on our fourth card we have 3 choices again similar to above.  On our last card we only have 44 choices left. So the probablity of getting a 2 pair should be (52*3*48*3*44)/(52 choose 5).  But this isn't correct.     Note: I edited this to "2 pair"; the original version was wrong. I would like to know how many switch can be stacked with fiber channel to get a best transfer rate between computers.  I have 200 computers connected to 12 3Com switches. All switches have two fiber ports.  Is the best way to have fiber backbone switch and connect fiber channel from every switch to the backbone? Or connect switch 1 to switch 2 to switch 3 to ... switch 12 to switch 1 again?  Thanks! Definite article — “on television” vs. “on the radio” Why are these different?        We heard the news on the radio.   We watched the news on television.      In this , the author says we must use television without the. Why? It makes me crazy. Is there a valid reason? Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$ I have a data set for a predictive model(predicting survival rate with certain acute medical condition on some animals) with 25 predictors where around 30% of the predictors are complete, 3 predictors are sitting at 30%,25% and 20% missing, and the others are missing at around 5% level. Around 50% of my data are complete cases. I'm new to dealing with missing values so I have a couple of questions regarding how to deal with it:   What can I do with the variable with 30% missing assuming it's MAR? Is 30% too high for imputation? What kinds of metrics can I use to make the decision between removal of the predictor, listwise removal, imputation or some other options? How should I deal with a predictor with 20% - 25% missing when I have reason to believe that it's MNAR? I'm thinking of using imputation on the remaining predictors with around 5% missingness. How do I make the decision on which imputation methods to use? Are they chosen on a case by case basis based on the individual predictors? How does one impute categorical values?  How does(should?) the imputation carry out in practice? Should it use complete cases or compute it iteratively somehow?  Does feature selection comes in before or after dealing with missing values? if at all? Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ Update: (Originally this was a comment, but I thought it was worth sharing here at the top.)   In the original question, I asked if there was a polite, socially-acceptable way to ask an Indian co-worker not to use the phrase "do the needful", as I didn't care for it.   In the years since I asked this question I've asked many people about the phrase. To the Indians I've asked in-person, it's not rude in any way. To Americans I've asked it varies anywhere from "I don't like it, but I don't mind it" to "It's very rude and makes me angry when I see it".   Through a long series of edits the question morphed and I wasn’t able to delete it because it had upvoted answers. The question, as it stands now, has no other answer.  I guess, though, if you're non-Indian and you find it rude, or if you're Indian and you've never realized someone might find it rude, this post may still have merit. Hopefully it does. It certainly has gotten a lot of views. Best wishes, folks.  End Update. See below for what's left of the actual question.    "Do the needful."   It's a phrase that I've only seen used in email, and I find it . . . presumptuous (maybe even rude). Regardless of prefacing with "please", one is commanding rather than asking for assistance.  I've only seen it used by those of Indian origin, so I've simply not mentioned it at all rather than worry about any cross-cultural offense that may come of bringing it up.  But still, I don't understand why it's used. Why not request rather than make two statements, one factual, one imperative?  For example, why would one use:        I'm told you have Jane's email address. Please do the needful.      instead of,        Would you send me Jane's email address? Why doesn't C3PO recognize Leia in R2's recorded message? Can an integrated HDMI port work both ways? Say I had a laptop [Correction: notebook, don't want my thighs medium rare], any OS, and it has an HDMI port built into the side. Would it be remotely possible to hack it into an input, provided it wasn't already designed for that? I would figure all video ports operate straight off the graphics chip, however, are there ways to override that functionality on the driver level? Driver hacks already exist for plenty of things such as the classic example: Bluetooth adapters (so we can play with our PS3 controllers and save a buck) However, is HDMI one of these tamperable interfaces? As far as the interface standard is concerned, are HDMI ports designed to be either way or are they Input/Output specific? Would a Cup of Tea Be Hotter If you Add the Milk Before or After Boiling Water? This is a bit of dispute between work colleagues. An answer would be greatly appreciated.  My argument is as follows:  If you add X amount of milk at a temperature of M to a mug at room temperature R before adding X amount of water at temp W, the result would be a cooler cup of tea than if you'd added the hot water first. This would be due to the milk reducing the overall temperature of the mug in the time that has been in it resulting in the addition of boiling water having less of an effect and resulting in a slightly cooler cup of tea/coffee.   Conversely, if you add the hot water first, due to the mug being heated, the effect of adding the cool milk will be less and therefore the end result will be a hotter cup of tea.  My Colleagues arguement is simply that it would not make a difference but without any form of justification.   If anyone can give real scientific light to this issue it would be greatly appreciated :) Is it a good practice to place C++ definitions in header files? My personal style with C++ has always to put class declarations in an include file, and definitions in a .cpp file, very much like stipulated in . Admittedly, part of the reason I like this style probably has to do with all the years I spent coding Modula-2 and Ada, both of which have a similar scheme with specification files and body files. I have a coworker, much more knowledgeable in C++ than I, who is insisting that all C++ declarations should, where possible, include the definitions right there in the header file. He's not saying this is a valid alternate style, or even a slightly better style, but rather this is the new universally-accepted style that everyone is now using for C++. I'm not as limber as I used to be, so I'm not really anxious to scrabble up onto this bandwagon of his until I see a few more people up there with him. So how common is this idiom really? Just to give some structure to the answers: Is it now The Way™, very common, somewhat common, uncommon, or bug-out crazy? The error message is :the grub-efi-amd64-signed package failed to install into /target/. Without the GRUB bootloader the system will not boot  I'm trying to install on HP pavilion g6-2303 TX using USB.  How do I fix this problem? wireless network adapter supported on ubuntu? Draw ellipses with different coloring for their intersection in tikz? How do I color intersection of two ellipses in different color. For instances, the ellipses should themselves be colored as yellow and their intersection as red. My code for drawing ellipses is given below  \draw (-4,0) ellipse (8 and 3); \draw (8,0) ellipse (8 and 3);   UPDATE: In the comments to individual answers, I have requested to show it for three ellipses. How to use numpad / on a keyboard without a numpad? I've just got into Blender, and I like it a lot. But as I just switched from a stationary computer to a MacBook, I've got some new problems regarding the numpad. I can use 1-9 perfectly, But I have no idea how to emulate buttons like / (used to enter local view). I've searched the web for an answer, but I can't find any. Generated Frattini Lemma Let $A$ be a normal subgroup of a group $G$. Let $B$ be a subgroup of   $A$ with the property that whenever $B_1 \leq A$ is conjugate to $B$   in $G$ then it is conjugate to $B$ in $A$. Prove that $G=AN_G(B)$   Help me some hints. How can I customize different nodepart with different textures? My question is how can I customize hatch patterns(grid, vertical line, etc.) for different nodepart in a node?  The following is a failed try.  \documentclass[border=2mm,convert]{standalone} \usepackage{tikz} \usetikzlibrary{shapes.multipart,patterns}  \begin{document} \begin{tikzpicture} \node[rectangle split,rectangle split parts =2,rectangle split part fill={red!30,red!10}, rectangle split part pattern={grid,},draw]{one \nodepart{second}two}; \end{tikzpicture} \end{document} What is a word for someone who is experienced and wise, but who deliberately acts naïve?    I don’t intend it for sarcastic use; I’m trying to describe someone succeeding in making people think that that person truly is naïve.   Is there a word for this? Take most recent (or concatenate) edit comments when editing during grace period Recently, I edited  to update a link to a better resource. I clicked "Save" without entering a description of the edit (edit #1).  Then I realised that the question was asked by a new user who may not be able to easily identify the change that was made. So I edited it again, this time only to add a detailed edit description (edit #2).  Since edit #2 was made during the grace period (within 5 minutes of edit #1), the system has merged them into a single edit, but only the edit comment from edit #1 was recorded in the revision history, whereas edit #2's comment was simply discarded/ignored.  I'd suggest that either the most recent (non-default) edit comment is taken or that the comments are concatenated to provide a more useful edit history Why would spacetime curvature cause gravity? It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force?  Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this?  When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place?