A combinatorics proof If $C(n, a)=C(n, b)$. Prove that $a=b$ or $n=a+b$. I looked for this on the internet. But no clue. I tried it myself, but failed. I have already asked for the help by posting similar question, about 20 days ago, and got some hints and finally, i was unable to complete the proof myself. The simplest proof is needed and will be appreciated.
Solving factorial equations I am given that $${n \choose a}={n \choose b}$$ I need to show that $n=a+b$ or $a=b$.  Now I am left with $a!(n-a)!=b!(n-b)!$.  I don't know what to do next. I checked there on google to solve the above, it says:- "On comparing, we get the required result." But it makes no sense to me.
I'm looking into using Named Credentials (spring '15) to simplify apex callouts to an external web service.  I actually thought, based on the documentation, that this would help, and I'm referring correctly to the Named Credential in my code and have also given my profile access to the NC, but I'm faced with this error when I do the callout:     System.UnexpectedException: You don't have permission to view this   data. Ask your administrator to set up authentication for the external   data source   This seems to say that NC are tied to the use of External Data Source, which is absolutely unrelated to what I want to do, and the documentation doesn't mention that they are a requirement.  Have any of you run into this error when testing out Named Credentials? Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? Cycle option when drawing between nodes in TikZ Cycle lets you close curves when drawing, and  \coordinate (a) at (0,0); \coordinate (b) at (1,0); \coordinate (c) at (0.5, -1); \draw (a) -- (b) -- (c) -- cycle;   works fine. Shouldn't also  \node (a) at (0,0) {}; \node (b) at (1,0) {}; \node (c) at (0.5, -1) {}; \draw (a) -- (b) -- (c) -- cycle;   give the same result?  The former gives a triangle, while the second doesn't close up. Is this a bug or am I missing something regarding the cycle option?  (Of course, as a workaround I could use the coordinates to draw the cycled path, and then draw the nodes at the same coordinates, but still...) Clicking community promotion ad renders an instance of SO within the frame of that ad Today I came across an ad on this question on SO:  (I don't expect it to stay very long).   There was a community promotion ad on the right hand side of the site, I think for one of our English sites. It had a sentence that I only partially remember:     Idiom ... poor or even poorer   I found it interesting, so I clicked that ad. Which caused a new instance of SO to open within the ad space. Unfortunately I failed to take a screenshot. I've been refreshing the page like crazy, but can't get the ad to appear again.  Edit: I managed to get a new community add, for the photography site, and captured a screenshot:   Edit2: This bug still persists, I saw it today with an OOP add from CodeReview, and an area 51 add. I'm wanting to use Xen Hypervisor to be able to run both Ubuntu 12.04 and Windows 7 at the same time using virtualization. I can find plenty of XCP tutorials but I'm not looking to install the cloud stack, I just want the server confined to my machine.  All I need is a link to a decent tutorial for this purpose. I've read a ton of articles that lead me to believe I'm on the right track, but not enough to feel comfortable getting it right if I just dive in. I'm getting caught up on whether I need to install Ubuntu and then install Xen into it, or do I need to install Xen on a small partition and then leave the rest of the hard drive open for the virtualization... I dunno.  Basically, I need the virtualization of Windows 7 for Adobe Premiere and After Effects (until Lightworks for Linux comes out and I can afford NUKEX). If I didn't need those, I'd be rocking out a straight up Ubuntu machine or just using VirtualBox for my virtualization. However, I need Windows to run as an HVM so it runs like I would expect rather than having the issues I may run into with VirtualBox (video drivers, etc).  So, any suggestions? Opt out of moderator election notifications Excel text alignment with custom formatting Is there anything in a custom formatted formula which defines the text alignment?  I know the different parts of the formula is numbers, negative numbers and so on, but I need to set the alignment.  Example:  _*# ##0_;_*(# ##0)_;_*"-"??_;_@_ Fatal error occurred, no output PDF file produced in php file We are using following code in .php script for generate .pdf file.  $result1 = exec("R CMD Sweave /var/www/html/r/r_latex_sweave.Rnw");  $result2 = exec("pdflatex /var/www/html/r/r_latex_sweave.tex");   but, we are getting this errors when we are executing .Rnw and .tex files.  ! I can't write on file `r_latex_sweave.log'. Please type another transcript file name:  ! Emergency stop !  ==> Fatal error occurred, no output PDF file produced!  Note: We have .tex and .Rnw folder in /var/www/html folder? How to delete duplicate LiDAR points? How can I delete duplicate LiDAR points? I incorrectly read the same las twice and saved as a one file resulting in duplicate XY points.  How should I delete these duplicate points? SSL certificate for a public IP address? I just tried buying a Comodo Positive SSL but it was rejected due to not supporting a public IP address, but instead they only support a domain name.  Does anyone know any SSL certificate provider that supports public IP address instead of a domain name?  My company has a dedicated server hosted with a web hosting company, which is used for running a bugtracker for multiple projects (for multiple clients). Since it's only used for a bugtracker, we don't need a domain name (our clients access it by typing the public IP in their browser). How could I get the terminal to be launched once I have logged in, so that typing commands is much simpler? I'm going to guess that the answer is "no" based on the below error message (and ), but is there anyway to perform a cross-database query using PostgreSQL?  databaseA=# select * from databaseB.public.someTableName; ERROR:  cross-database references are not implemented:  "databaseB.public.someTableName"   I'm working with some data that is partitioned across two databases although data is really shared between the two (userid columns in one database come from the users table in the other database). I have no idea why these are two separate databases instead of schema, but c'est la vie... Drafts for edits get lost on leaving the Android app When editing in the Stack Exchange Android app and navigating away to get some reference URLs the text is gone when I go back into the app.  Seems like the edit isn't kept in memory. Can you please save them in memory? (drafts) Conjugate subgroup strictly contained in the initial subgroup? Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$?   If $H$ has finite index or finite order, this is not possible. "How to [...]?" and "Where to [...]?" Questions that are not questions. Is this defensible? Adding a question mark to the end of a "How to" or "Where to" sentence appears to be quite common. Here are two examples from this very site:  How to punctuate a list of questions? ()  Where to put the periods when using a parenthetical sentence? ()  In my mind, the question mark doesn't belong at all, and it bothers me.  Am I being unreasonable? Is this practice defensible? Moving an already-running process to Screen My office job routinely sees me connected to a Linux box via VNC.  Sometimes I start a remote job on the console, and realize later that it runs much longer than expected. (Should have started that one under Screen in the first place...)  I don't want to keep my workstation running overnight just to keep the VNC session open; I would like  to move that already-running remote job into a Screen session (on the remote box), so I can power down the workstation (and reconnect next morning).  How can this be done, if at all? Why aren't search engines indexing my content? This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes? I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$.   I'm trying to figure out if they're isomorphic (as rings I suppose) or not for primes $p=2,5,11$.   I don't think they are for $p=11$, since $x^2-2$ is irreducible over $\mathbb{F}_{11}$, so $R_1$ is a field. But $x^2-3$ has $x=5,6$ as solution in $\mathbb{F}_{11}$, so $R_2$ is not even a domain.  For $p=5$, both polynomials are irreducible, so both rings are fields with $25$ elements.  I know from my previous studies that any two finite fields of the same order are isomorphic, but I'm curious if there is a simpler way to show the isomorphism in this case, without resorting to that theorem.  For $p=2$, neither ring is even a domain as $x=0$ is a solution of $x^2-2$ and $x=1$ is a solution for $x^2-3$, but I'm not sure how to proceed after that. Thank you for any help. Does construction of a Lightsaber require use of the Force? It's true that Lightsaber construction is a mandatory skill for a trained Force user, but I am not sure about it in reverse.  @DVK has commented on  that:     Grievous wasn't a Jedi (or a Force Sensitive) at all. He couldn't construct one.   I always thought that a Lightsaber was a normal piece of technology which even non-Force users could construct, and you only need Force sensitivity to master  Lightsaber skills. But, I could be wrong.  Does construction of a Lightsaber require use of the Force? If yes, what role does the Force play in the construction of a Lightsaber? I'm working on this problem already, the domain and co-domain are the set of Rational Numbers. we see that setting $x = 1$ and $y = n$, we obtain $f(n + 1) = f(n) + 1$  But I'm trying to prove that $f(n) = n + 1$ for all $n$, the case when $n = 1, 2$ is obviously true, how do I approach this via induction?