How many solutions are there to the inequality $x_1 + x_2 + x_3 ≤ 11$, where $x_1, x_2$ and $x_3$ are non-negative integers? [Hint: Introduce an auxiliary variable $x_4$ such that $x_1 + x_2 + x_3$ +  $x_4$ = 11.]  Would my reasoning be correct if I let $x_4 = x_1 + x_2 + x_3, x_4 = 11$  Then proceeded as normally with $14\choose11$? I'm a bit unsure of how the auxiliary variable comes to play.
How can I compute the number of lists of the form $(a_1,\ldots,a_k)$, where $a_i \geq 0$ are integers and $\sum a_i \leq n$? What is the easiest way to derive the answer?
I am no native speaker and always confused about the comma in introductory phrases, in particular in prepositional phrases. Is there any hard rule when a comma must be set? If I make a google search for certain phrases, I often find both variants. Typical examples where I am not sure whether a comma must be set are:        First[ly][,] I must phone mother   In this case[,] we must...   For simple problems[,] the algorithm...   From Lemma 1.2[,] we obtain...   For a typical user[,] the algorithm...   In our theory[,] we...   A dark line on the horizon[,] the mountain range we were headed for seemed somehow not as threatening as when we had started out on the trip.   Avidly reading her novel[,] Jane did not notice that her stop was approaching.   To be sure[,] it is not easy to see a solution.   When you get to Rome[,] give me a ring. How do you access the matched groups in a JavaScript regular expression? I want to match a portion of a string using a  and then access that parenthesized substring:  var myString = "something format_abc"; // I want "abc"  var arr = /(?:^|\s)format_(.*?)(?:\s|$)/.exec(myString);  console.log(arr);     // Prints: [" format_abc", "abc"] .. so far so good. console.log(arr[1]);  // Prints: undefined  (???) console.log(arr[0]);  // Prints: format_undefined (!!!)   What am I doing wrong?    I've discovered that there was nothing wrong with the regular expression code above: the actual string which I was testing against was this:  "date format_%A"   Reporting that "%A" is undefined seems a very strange behaviour, but it is not directly related to this question, so I've opened a new one, .    The issue was that console.log takes its parameters like a printf statement, and since the string I was logging ("%A") had a special value, it was trying to find the value of the next parameter. How do I set my level, unlocked weapon equip slots and playthrough mode? why could a check if (document.addEventListener) returns false bcmwl-kernel-source broken on kernel: 5.8.0-34-generic How to prove that the set of rational numbers are countable? What are the differences between an Ubuntu Long Term Support release (LTS) and a normal release? I read the following dev email  and when i read this part:  Note that this does not completely remove the ability for extensions (e.g. Dropbox) to add emblems programmatically by using the libnautilus-extension library, just the user-added emblems from the properties dialog/emblems sidebar.   There is ability to bring emblems programmatically. So, How can i use libnautilus-extension library to add some emblems like Dropbox does? Or is there any tweak tool for this? Will Windows 10's reset delete my Linux partitions? I want to use the "reset my PC" feature of Windows 10 to wipe the data on my Windows partition. But I have also a Linux system installed, which I want to keep.  Will Windows 10's full reset wipe all my disk, or just Windows's partitions ? Why would someone buy a way out-of-the-money call option that's expiring soon? I swing trade equity options every once in a while for fun. This morning I witnessed something I do not understand at all.  In the attached photo, why would someone purchase 6 contracts for .05 at $76.  I just can't wrap my head around this!    Scenario:   Underlying Asset: RY.TSX  Current Price: $71.70 Days to expiry: 4 Days     EDIT: It happened again the next day with 74, see new table with headers as per request:   My only reason for asking is because RY.TSX is the Royal Bank of Canada, this stock will not move multiple dollars in 3 days(expiry in this scenario).  Why buy 10 @ 74 for .04 each instead of just buying 10 @73 for .04 which would have gone through.  If you do the math(excluding commissions): .04 * 100(shares in a contract) * 10 contracts = $40 Stock moves to 74 and they are in the money lets say for .20 .20*100*10 = $200  I just don't see the reward as 99% of the time you will lose this wager, it is just not worth it even if it does happen once in a blue moon.  I can think of much better things to do with $40.  For the skeptics another screen shot to actually confirm this order: How does carbon dioxide or water vapour absorb thermal infra red radiation from the sun? We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2? Is there a unique solution for this quadratic matrix equation? Opposite of "straight talk" What is the opposite for the straight talk idiom? How do I best call the activity when someone makes a very long preamble before he says what he wants? How to create a link to a URL provided in a bib file with custom text Using biblatex, how can I get a URL from the bib file so that I can use it in the \href command?  I am guessing that the \usefield command is what I am looking for and that in order to use that I would need the \entrydata command. But both are undefined.  \documentclass{article} \usepackage[backend=biber]{biblatex} \usepackage{hyperref}  \usepackage{filecontents} \begin{filecontents*}{sources.bib} @online{biblatex,     url = {http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf},     urldate = {2018-02-02}, } \end{filecontents*}  \addbibresource{sources.bib}  \begin{document}      % This would work, but I don't want to duplicate the URL from the bib file:     \href{http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf}{biblatex documentation}      % Does not work because \citeurl{biblatex} is already a link, not the URL:     %The \href{\citeurl{biblatex}}{biblatex documentation}      % Does not work because \usefield and \entrydata are undefined:     %The \entrydata{biblatex}{\usefield\href{url}{biblatex documentation}}      \printbibliography \end{document} I'm trying to compute the difference between the output of two awk commands but my simple attempts at it seem to be failing. Here is what I'm trying:  diff $(awk '{print $3}' f1.txt | sort -u) $(awk '{print $2}' f2.txt | sort -u)   This doesn't work for reasons unknown to me. I was under the assumption that $() construct was used to capture the output of another command but my "diff" invocation fails to recognize the two inputs given to it. Is there any way I can make this work.  By the way, I can't use the obvious solution of writing the output of those two commands to separate files given that I'm logged on to a production box with no 'write' privileges. Most efficient way to test equality of lambda expressions I upgraded from 17.04 to 17.10 on a Lenovo T540p which has broken two finger touchpad scrolling and right clicking Trackpad two finger scrolling and right-click issue with Ubuntu 17.10 (i386, 64bit, GNOME 3.26.1) on a Lenovo T540p (the model with a Intel Core i5-4210M Haswell CPU etc) Is it possible to call a non-exported function that resides in an exe? I get that some spells are "especially swift" to cast and thus have a casting time of "Bonus Action". But is there any reason I couldn't cast such a spell as my Action?   Does it make a difference if the idea is for the one spell to be cast all by itself on a turn or if the desire is to use another Bonus Action (possibly a spell) on the same turn? Centering figure captions in IEEEtran I am working on a paper using the IEEEtran template in LyX.  I have run into a problem: While the contents (in this case, a table) of the figure float are centered, the caption is aligned to the left (see screenshot below). How can I center the caption as well?    This is generated by the following LyX document:    What I've tried:   Center the table (the table is centered) Center all the contents in the float (only the table is centered) Select the caption and center it (no effect)   Technical details:   LyX 2.0.5.1 Mac OS X 10.7.5 (Lion) All centering was performed by Right Click>Paragraph Settings>Align: Center