How to create Pascal's triangle like this:
I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code  \begin{tikzpicture} \foreach \n in {0,...,4} {   \foreach \k in {0,...,\n} {     \node at (\k-\n/2,-\n) {${\n \choose \k}$};   } } \end{tikzpicture}   The result is this   Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?  The result should look like this:
I am currently studying  answer on the Moore-Penrose pseudoinverse and the Euclidean norm by the user &quot;Etienne dM&quot;. The first point of their answer proceeds as follows:  Let $x$ be $A^+y$.  Let me begin by the second point. For all $z$, we have: \begin{align} \lVert Az-b \rVert_2^2 &amp;= \lVert Ax-b \rVert_2^2 + \lVert A(z-x) \rVert_2^2 + 2 (z-x)^TA^T(Ax-b)\\ &amp; \geq  \lVert Ax-b \rVert_2^2 + 2 (z-x)^TA^T(Ax-b) \end{align} Moreover, because $(AA^+)^T = AA^+$, $$ A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$$ Thus, we prove that for all $z$, $\rVert Az-b \lVert_2^2 \geq\rVert Ax-b \lVert_2^2$, that is to say $A^+b$ is as close as possible to $y$ in term of the Euclidian norm $\lVert Ax-b\rVert_2$.   I realise that $x = A^+ y$, but I don't understand how any of this implies that &quot;$A^+ b$ is as close as possible to $y$ in terms of the Euclidean norm $\lVert Ax-b\rVert_2$&quot;. I would greatly appreciate it if people would please take the time to explain this to me. The outlet for my range hood was misplaced. It needs to be 4 inches to the side. I can make the junction and move the wiring to the new location but I really don't want to have the empty plate exposed.   The wire is not long enough to reach the required location, so I will make a junction in the existing box, and add a new box at the correct location.  Is there any acceptable circumstance where I could tile over the "old" in-wall junction box?  I'm guessing not, but I figured I'd ask.  EDIT: I should clarify, based on the answers below... I know for a fact that this outlet is the only line on this entire circuit, all the way back to the panel. I am asking explicitly if there is any way to acceptably splice the wiring without exposing a junction box. I want to do something like:  MyObject myObj = GetMyObj(); // Create and fill a new object MyObject newObj = myObj.Clone();   And then make changes to the new object that are not reflected in the original object.  I don't often need this functionality, so when it's been necessary, I've resorted to creating a new object and then copying each property individually, but it always leaves me with the feeling that there is a better or more elegant way of handling the situation.  How can I clone or deep copy an object so that the cloned object can be modified without any changes being reflected in the original object? Create a trigger on all the last_modified columns in PostgreSQL In PostgreSQL 9.5, I have tables with columns in the form   prefix_last_modified timestamp without time zone NOT NULL DEFAULT  (clock_timestamp() AT TIME ZONE 'UTC')   I was looking for a way to set the last modified value automatically updated at each update of the rows, and I found  that defined the function:  CREATE OR REPLACE FUNCTION update_modified_column()  RETURNS TRIGGER AS $$ BEGIN     NEW.modified = now();     RETURN NEW;  END; $$ language 'plpgsql';   Now, I'd like to know if there is any way to pass the column name to the PostgreSQL function and to execute it to the NEW row? E.g.  CREATE OR REPLACE FUNCTION update_modified_column(varchar column)        RETURNS TRIGGER AS $$     BEGIN         NEW.column = now();         RETURN NEW;      END; $$ language 'plpgsql'; How to delete column/field in PyQGIS? Why can't variable names start with numbers? When I render a page using the Django template renderer, I can pass in a dictionary variable containing various values to manipulate them in the page using {{ myVar }}.  Is there a way to access the same variable in Javascript (perhaps using the DOM, I don't know how Django makes the variables accessible)? I want to be able to lookup details using an AJAX lookup based on the values contained in the variables passed in. Using a naked black hole as a mirror? The YouTube video is a good example how the gravity of this merging binary black holes bend light around themselves. I am foreign graduate student in USA. I recently completed my undergrad (from USA) and have a bit of money saved up. I know that F1 students cannot work off campus and other business activities such stock trading etc are prohibited as well. I wanted to know if opening an account in an investment management firm like Charles Schwabb or Betterment is allowed. I would not be working but I will still be earning a small interest on my deposits. If that is not allowed, are there any good and legal ways for foreign students to earn a little interest on savings? I stupidly ran sudo chown -R carlos:carlos / and carlos is my user. This changed all files to be Carlos and I could not boot up again (because the root files were now owned by #1000) and I cant get sudo acess in live cd (because the owner of /usr was changed to my user). Can anyone help me? FYI I have mounted my hard drive. I don't really get why these hyperbolic functions are named after trig functions. Can someone enlighten me? What do I do? I looked at a similar question to mine and there was an answer saying to make sure Java is allowed past the firewall. So I checked and it already is. I've tried uninstalling and installing both Minecraft and Java but nothing has changed.  says that all services are fine, so what's going on? I bought a 2nd account to use as an alternative in-game account and now I can't log in.  (I cleared the email and password boxes for obvious reasons in this screenshot: security purposes) Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs How to capture the scale of a scene? My framing in the following photo completely failed to capture the sheer height of the waterfall.  If you don't know I was standing at the top of the waterfall, and the "splinters" at the bottom are full-size tree trunks 2 feet in diameter, you wouldn't have a clue how tall it is.  It just looks like a smaller, shorter waterfall shot with a slower shutter speed.    My question isn't about this picture or waterfalls in particular, but about framing to capture the scale of your subject in general.  What are techniques to show just how big something really is?  Or conversely, something really small?  Especially things like rocks or water that look roughly the same at any size. How to access a usb flash drive from the terminal? I only have access to the command line. I need to backup my data (on the user folder) to a pen (USB DOK).    How can I mount a flash drive manually? What should be the copy command? How can I get CSS comments with jQuery? I am wondering how I can read CSS comments out of a linked stylesheet.  I have this sample CSS loaded via:  &lt;link rel="stylesheet" type="text/css" media="all" href="test.css" /&gt;   &nbsp;  #test1{ border:1px solid #000; } #test2{ border:1px solid #000; } #test3{/* sample comment text I'm trying to read */}   I'm testing this in FF3. The following javascript reads the rules but doesn't read the comments in#test3.  window.onload = function(){     s=document.styleSheets;     for(i=0;i &lt; s[0].cssRules.length;i++){         alert(s[0].cssRules[i].cssText);     } } When do I need to invoke \phantomsection? Cancellation of addition on convex sets I  found a question about a property of the . However the question was not properly answered (it used a projection argument which might not be true in a general Banach space).  I was wondering whether the following (weaker) statement holds:  Let $X$ be a Banach space and suppose $A,B,C_0\subset X$ are bounded, closed, convex and non-empty subset. Do we then have $$A+C_0=B+C_0\implies A=B?$$ Until now, I have been compiling my LaTeX documents with pdfLaTeX. My standard document preamble includes the "usual suspects" for non-English speakers (, , ) and a bunch of other packages. Assuming I were to convert to XeLaTeX, what modifications of my preamble are advisable? I'm aware that the expansion=true option of the  package is not (yet) available for XeLaTeX, and that  is sort of a default package for OpenType fonts. What other packages (and package options) should be removed and, vice versa, incorporated when switching from pdfLaTeX to XeLaTeX? Line/word wrapping in revision history and responses tab (and probably other places) and        ()   I just noticed that change. I didn't find new reports under .  Note the names and badge count in the first screenshot, and the badge count in the second one.  PS I can't link to the responses tab as it shows only to the profile owner.