I am trying an assignment problem  which I am unable to think about.      Question is -> How many $4$-subsets of $S= \{1,2, \ldots,20\}$ contain no consecutive integers?    I don't have any idea how to solve it.   Can someone please explain how to think about it ?
I am a newbie in combinatorics and still don't have enough tools to handle this kind of problems.  Assuming I have a set of integers: $ \{1,2,3,4,\dots,n\} $  In how many ways, can I choose $k$ numbers out of those $n$, such that none of them are consecutive?  For instance, for the following set $ \{1,2,3,4,5\} $  For $ k=3 $ I have only 1 option: $\{1,3,5\}$  My purpose is first of all to understand the way you would think about this  problem, not necessarily the solution itself (I have the final answer for it).  Thanks.
Every countably generated sigma algebra is generated by a random variable Controlling bandwith while using GPRS on Windows 7 I have an issue with my windows 7 machine using a lot of bandwith while connected to the internet. I'm trying to limit the bandwith by using Opera in Turbo mode, but some other program is sucking down content and emptying my GPRS mobile plan.   Is there some way I can control which applications are allowed to use bandwith while on GPRS? Preferable only Opera should be able to use the network.   Thanks. How to set a font for math mode? I would like to have something like Segoe UI of something similar. Thank you  \documentclass{article} \usepackage{fontspec} \usepackage{firamath-otf} \begin{document} \begin{equation} x+y=45 \end{equation}  \end{document}   Error: If you open any of localized meta site in browser's private mode, e.g.  you will see untranslated text on the top right corner:        This message is not found in Transifex, hence unable to localize.  Related bug report on ruSO.meta: mkdir error: no such file or directory on a linux machine running CENTOS v 7.8.2003 I am in a directory Neutrinos I now do  mkdir /pionloop/ this works. I then go to into this new directory pionloop and do mkdir /E_0.3 and this works as well. I thus have as a result a directory Neutrinos/pionloop/E_0.3.  Now, starting in Neutrinos I want to do this in one command and therefore do mkdir /pionloop/E_0.3 and get: `mkdir: cannot create directory ‘/pionloop/E_3.0’: No such file or directory  What is going wrong here?? Why does the command shuf file > file leave an empty file, but similar commands do not? I know this is sort of a duplicate of another question () but I wanted to expand on the question in response to the answers given.  The command  shuf example.txt &gt; example.txt  Returns a blank file, because the shell truncates the file before shuffling it, leaving only a blank file to shuffle. However,  cat example.txt | shuf &gt; example.txt  will produce a shuffled file as expected.  Why does the pipeline method work when the simple redirection doesn't? If the file is truncated before the commands are run, shouldn't the second method also leave an empty file? Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$ failed to fetch security undermined error [IP: 91.189.91.38 80] in ubuntu 20.04 convert ceil to floor How can I get bash to perform tab-completion for my aliases? I have a bunch of bash completion scripts set up (mostly using bash-it and some manually setup).    I also have a bunch of aliases setup for common tasks like gco for git checkout. Right now I can type git checkout dTab and develop is completed for me but when I type gco dTab it does not complete.    I'm assuming this is because the completion script is completing on git and it fails to see gco.    Is there a way to generically/programmatically get all of my completion scripts to work with my aliases?  Not being able to complete when using the alias kind of defeats the purpose of the alias. LaTeX to plain text for e.g. generation of statistics I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise. So, i want to make a timer that will turn on/off a circuit. I want to turn on the connected device at 08:00 and stop it at 20:00, so the interval is 12 hours. Correct me if i'm wrong but i can't use IC 555(or another IC), i'll need some huge resistors/capacitors. Is there another way to build that other than using a microcontroller? Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$? Let $f$ be a linear functional on a normed space $(X, \|\cdot\|)$. Prove that $f$ is continuous at $x_0=0$ if and only if $f$ is continuous at every $x\in X$.  I understand that the $\Leftarrow$ is trivial but what about the other way? No CUDA and GPU option What are Java command line options to set to allow JVM to be remotely debugged? I know there's some JAVA_OPTS to set to remotely debug a Java program.  What are they and what do they mean ? What is the difference between Non-preemptive, Preemptive and Selective Preemptive Kernel? Linux Debian codenames I've got a few, quite silly, non-technical questions about giving  to  releases.  Each Debian release has its unique codename, which is (so far) a characters' name from  movies by .  Here is list of all assigned codenames so far:   release 1.1 is  (Buzz Lightyear) - the spaceman, release 1.2 is  - the tyrannosaurus, release 1.3.x is  (Bo Peep) - the girl who took care of the sheep, release 2.0 is  - the piggy bank, release 2.1 is  (Slinky Dog) - the toy dog, release 2.2 is  - Mr. Potato, release 3.0 is  -  the cowboy, release 3.1 is  - the sergeant of the Green Plastic Army Men, release 4.0 is  - the toy blackboard (Etch-a-Sketch), release 5.0 is  - the toy binoculars, release 6.0 is  - the name for the three-eyed aliens, release 7.0 is  - the name of the rubber toy penguin with a red bow tie, release 8.0 is  - the name of the yodelling cowgirl, release 9.0 is  - a purple rubbery octopus toy at , release 10.0 is  - 's pet dachshund (currently ).   List of upcoming major Debian releases' codenames after :   release 11.0 is  - 's horse (currently ), release 12.0 is  - intelligent worm toy with a built-in flashlight.   There are also:   special codename  () which is symbolic link to codename which is currently unstable,  which is symbolic link to codename which is currently stable,  which is symbolic link to codename which is currently testing.   The list of Toy Story  is quite  but at some time, there will be no more characters' names to assign.  My questions are:   What codenames will be assigned if we run out of characters' names? Who decides what is codename of next release (please don't answer ambiguously like: 'community')? How many releases' names are planned ahead?   BTW: Interesting quote from :     The  of using Toy Story names was  by  who   was, at the time, the Debian Project Leader and was working also at   , the company that produced the movies.     by  (: ). Lag compensation with networked 2D games I want to make a 2D game that is basically a physics driven sandbox / activity game. There is something I really do not understand though. From research, it seems like updates from the server should only be about every 100ms. I can see how this works for a player since they can just concurrently simulate physics and do lag compensation through interpolation.  What I do not understand is how this works for updates from other players. If clients only get notified of player positions every 100ms, I do not see how that works because a lot can happen in 100ms. The player could have changed direction twice or so in that time. I was wondering if anyone would have some insight on this issue.  Basically how does this work for shooting and stuff like that?  Thanks Storing datetime values in Shapefiles using QGIS? I have a shapefile with date in a string field. I create a new field with date type and use the calculator to make the conversion. But the result only provide the date part, the time part is removed. Even when the preview show the correct conversion.  Here you can see the original string field time and the result date time2. And the use of the conversion function to_datetime() and the preview result at the end    If you go to the layer properties you can change the edit mode to date/time. Then if go to the data and try to edit time2 you get the current date/time. So is possible save date and time in the same field. But the update tool still doesn't work as expected      As J.R said. Even when you can change QGIS edit to allow Date/Time the save to shapefile will discard the time. I'm trying to find a way to automatically clean/delete the temporary files from C:\Documents and Settings\user\Local Settings\Temp\ at shutdown/boot on my Windows XP Professional x64 SP2 machine.  The main reason for this is to clean Local Settings/Temp to remove old junk automatically. (I had thought I had this covered through Firefox settings, but I was only clearing temporary Internet files). Currently, there's around 3&nbsp;GB of temporary files, and I'd like to prevent this build up without having to remember to do anything on my part.  I have already looked at using  (explained in Super User question ), but it doesn't seem to touch that directory.  I'm aware of programs such as , but I would ideally like to use a solution that doesn't require additional software (so, I expect, something using a scheduled BAT/CMD script would be what I'm looking for).  One of the potential issues, I believe, is that there may be legitimate files in the Temp folder that may be needed on the next boot (for software installations or updates, for example) and the perfect solution would be able to distinguish those and leave them until they weren't needed (Note: I realise this perfect solution may not be reasonably possible).  I've already set up Disk Cleaner on a shutdown script, but when cleaning temporary files, Disk Cleaner doesn't clean this particular directory, and so I'm looking for alternative options (see Super User question  for a little more detail).  So, what are my options for automatically deleting the temp files from C:\Documents and Settings\user\Local Settings\Temp?