Finding maximum value of ${n \choose r}$ for given value of n
How do you prove ${n \choose k}$ is maximum when $k$ is $ \lceil \tfrac n2 \rceil$ or $ \lfloor \tfrac n2\rfloor $?
As a native English speaker, which two further languages would give me the most travel utility worldwide? This follows on from a travel conversation a group of us had in a hostel in London.  Assuming I am travelling randomly between countries and continents, which two languages in addition to English would provide the most usefulness in any given country?  I suspect Spanish (North America, South America) and French (Europe, Africa) but with the base understanding between French and Spanish maybe the time would be best spent learning another language?  This question is a real puzzler since the following constraints apply   Length of time required to be "functional" in the language (cost-time trade off) Chance of reaching proficiency within any meaningful timeframe (Mandarin) Regional dialects destroying proficiency (Arabic) Easier Secondary languages underpinning the society (Arabic versus French argument) Similarity of Language families (Romance) leading to wasted effort (Italian v French) Utility of the Language (German versus English when English is understood) Enjoyment is a minor consideration   Happy to consider any other thoughts but it is a really interesting topic for consideration if you are linguistically minded. I have some survey response data that are inconsistent and I'm not sure what is the best way to handle them.  This is the nature of the inconsistency:    Have you ever had a food poisoning episode that you think might have been caused by the food in a particular restaurant ? a) Yes b) No Did any of these episodes prevent you from eating at that restaurant again?  a) Does not apply, never have had a food poisoning episode b) Yes c) No Following these episodes, did you come back to your normal condition after 24 hours?  a) Does not apply, never have had a food poisoning episode b) Yes c) No   I believe that the intent of the designers of the survey was that anyone answering "No" to 1) should answer "Does not apply" to 2) and 3), but that's not how it worked. A large group of respondents picked "No" for 1) and "No" for at least one of 2) or 3).    What is the best way to deal with this inconsistency?    Where are some good references (books, papers, websites, etc.) to look at discussions and solutions for issues of this kind, and data cleaning in general? I want to collect the most common mistakes, misconceptions, pitfalls, etc that (La)TeX and Friends users make. Please one answer for each mistake, misconception, pitfall, etc.  I am only interested in more technical, objective cases rather than psychological, social, subjective cases.  From this complete infinite list of technical and objective cases, we can have a reference how  to teach newbies much better directly or indirectly via writing a good book.  The technical and object cases are, for example, as follows.   Beginners sometimes use $\huge E=mc^2$ with the hope they will get a huge formula. It actually does not produce the expected result. The correct way is \huge $E=mc^2$. Beginner sometimes use longtable inside table because they assume longtable is the longer version of tabular which is able to be sandwiched in table. etc. OpenLayers 3 WMS GetFeatureInfo popup I have a map a at the link below:    I added one of my GeoServer WMS layers to the map, now I try to add the attributes window pop up on click. I found to examples on the OpenLayers 3 homepage, but I cannot put together a working code...      Is there any chance to have a working sample somewhere? How do I find this conditional expectation? How to install LXDM instead of GDM3? I'm running Debian Wheezy and my default Display Manager is GDM3, and I would like to replace it with LXDM. The problem is that I'm trying to run # apt-get install lxde and it's not working, what am I doing wrong?  My /etc/apt/sources.list is:  deb cdrom:[Debian GNU/Linux testing _Wheezy_ - Official Snapshot i386 DVD Binar$  deb http://ftp.br.debian.org/debian wheezy main deb-src http://ftp.br.debian.org/debian/ wheezy main  deb http://security.debian.org/ wheezy/updates main contrib deb-src http://security.debian.org/ wheezy/updates main contrib How do I install Netbeans? I used NetBeans for Java development on Windows and remember finding it in the Ubuntu Software Center on Natty, but now it's gone. Is there a PPA I can use or will it be re-added to the official repos? Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$ Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. How to navigate to C drive in bash on WSL-Ubuntu? Suddenly unable to screenshot- Do not have permission to save files in the location where screenshots are stored Recieved the error:  'Your screenshot could not be saved. You don't have permission to save files in the location where screenshots are stored.' with OS X Yosemite.  I opened terminal and tried directing the screenshots to save on my desktop:  Brigittes-Air:~ eggsalad$ defaults read com.apple.screencapture location 2017-01-06 03:04:47.765 defaults[17729:1947540]  The domain/default pair of (com.apple.screencapture, location) does not exist  So I tried writing and:  Brigittes-Air:~ eggsalad$ defaults write com.apple.screencapture location ~/Desktop 2017-01-06 03:05:08.652 defaults[17742:1948451] Could not write domain com.apple.screencapture; exiting  I'm not sure where the screenshots are being stored if the location does not exist. Ideas?  EDIT  I ran something else and got a different response, but defaults are still unchanged.   Brigittes-Air:~ eggsalad$ defaults write com.apple.screencapture location/Users/eggsalad/Desktop killall SystemUIServer  2017-01-06 03:22:35.444 defaults[18265:2001767] Unexpected argument SystemUIServer; leaving defaults unchanged.  2nd EDIT  Result of:  ls -l ~/Library/Preferences/com.apple.screencapture.plist  was:   ls: /Users/eggsalad/Library/Preferences/com.apple.screencapture.plist: No such file or directory Securing SSH server against bruteforcing I have a little SVN server, old dell optiplex running debian. I don't have that high demands on my server, because its just a little SVN server... but do want it to be secure.  I just renewed my server to a newer and better optiplex, and started looking a bit into the old server. I took it down after experiencing problems. When I check the logs, its full of brute-force attempts and somehow someone has succeeded to enter my machine. This person created some extra volume called "knarkgosse" with two dirs "root" and "swap1" or something.  Don't really know why and what they do, but sure do want to prevent this from happening again. I find this a bit strange though because I change my password ever few months or so, and the passwords are always random letters and numbers put together... not easy to brute-force.  I know I can prevent root from logging in, and use sudoers... and change the SSH port, but what more can I do?  So I have a few questions:   How can I prevent logging in for 5 minutes after X amount of incorrect tries. Or slow tries down after each incorrect try? Is there some kind of central blacklist which a server can connect to? A blacklist that keeps track of IP addresses that are "unsafe" and should never be granted access? What more can I do to apply safety to my server?   Like I said earlier, I am running Debian 5 with Apache (www-data user problem?), svn, mysql, php, phpmyadmin, hudson. It is on a home network with port forwarding on 80, 443, 8080, 8180, 23 and 22. Flag dialogue popup is stacked behind realtime-post-deleted-notification banners There is a little bug that becomes visible when a page contains realtime-post-deleted-notification banners (that's the CSS classname of these red banners appearing on answers which just got deleted, see screenshot below) and you try to open the flag dialogue above them:    Obviously, the dialogue appears below the banners, which it clearly should not do. It also has no input focus, trying to interact with the dialogue fails and either closes it (clicking outside of the red banners) or reloads the page (clicking on the red banners).  The cause for this is simply the z-index ordering in the site's CSS. Relevant CSS snippets:  .realtime-post-deleted-notification {     position: absolute;     text-align: center;     z-index: 100; }  .popup {     display: none;     padding: 15px;     position: absolute;     z-index: 10; }   So the fix for this minor bug would probably be to raise the z-index of the popup class to something greater than 100. state that "at best" is an idiom. But, what is the grammatical function of "at best" (for example, in the below sentences?)        Their response to the proposal was, at best, cool.   The government's response seems to have been at best confused and at   worst dishonest.   If he drops the course now, at best he’ll get an incomplete, and he could fail.      Thanks Factorial of 0 - a convenience? How are dashes implemented in TeX? How does LaTeX implement the differences between the dashes?  -, --, and --- all print different dashes, but - is not an active character: it has catcode 12, the same as digits like 2. How to filter a list by hour I have a really strange situation here. My PC works fine, at least in most cases, but there's one thing that I can't deal with. When I try to copy a file from my pendrive, everything is ok -- I got 16-19M/s , it works pretty well. But when I try to copy something to the same pendrive, my PC freezes. The mouse pointer stops moving for a sec or two, then it moves a little bit and it stops again. When something is playing, for example, in Amarok, the sound acts like a machine gun. The speed jumps from 500K/s to 15M/s, average 8M/s. This occurs only when I'm copying something to a pendrive. When the process of copying is done, everything backs to normal.   I tried everything -- other pendrive, a different USB port on front panel or those ports from back, I even changed the USB pins on motherboard (front panel), but no matter where I put my USB stick, it's always the same. I tried different filesystem -- fat32, ext4. I have no problem with the device on Windows, on my laptop. It has to be my PC or something in my system. I have no idea what to look for. I'm using Debian testing with standalone Openbox. My PC is kind of old -- Pentium D 3GHz, 1GiB of RAM, 1,5TB WD Green disk. If you have something that would help me to solve this issue, I'd be glad to hear that.  I don't know what else info I should provide, but if you need something, just ask, I'll update this post as soon as possible.  I tried to reproduce this problem on ubuntu 13.04 live cd. I mounted my encrypted partition + encrypted swap and connected my pendrive to a usb port. Next I tried to start some apps, and now I have ~820MiB in RAM and about 400MiB in SWAP. There's no problem with copying, no freezing at all, everything is as it should be. So, it looks like it's a fault of the system, but where exactly? What would cause such a weird behavior? Casting a Somatic, non-Material spell while holding a focus In  a side issue came up of whether you can cast a spell with Somatic components but no Material components while holding a focus in the hand used for Somatic components.  My reading of the rules sees "A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus--but it can be the same hand that he or she uses to perform somatic components." - it doesn't say you have to be holding the focus for the purpose of avoiding materials.  However, @Adeptus and @enkryptor argue that - since that rule is in the section on Material components - you don't consult it in for spells without Material, and fall back to the Somatic rule "the caster must have free use of at least one hand".  (I also appeal to the logic of fluff - why should it be harder to cast a spell with fewer components?)  Is there an official RAW or Word Of God on this matter? Can you cast a material-less, somatic spell while holding a focus in your only free hand? I just learned that using CBC encryption with an IV which is predictable is not secure.  From what I understand, using certain plain texts, and then guessing the IV that it uses, the attacker can verify if the IV he guessed was right. How does this qualify as an attack? How to delete all files in a current directory starting with a dot?