hi guys can you help me with this? Proof that expression is integer $$\frac{(2n)!}{2^nn!}$$
Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$.   Could anyone please help me with this proving? Thanks!
I've been using Ubuntu for a few years now, but I just installed Version 16.04, and I've been having a lot of issues. I'm not a command line expert, but I tried to install some software last night, and it is creating issues (and also not working properly).   When I run sudo apt-get -f install, I see the following message:  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following additional packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody Suggested packages:   lua-zlib lua-dbi-postgresql lua-dbi-mysql lua-dbi-sqlite3 The following packages will be REMOVED:   jitsi-meet-tokens The following NEW packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody 0 upgraded, 7 newly installed, 1 to remove and 22 not upgraded. 3 not fully installed or removed. Need to get 0 B/413 kB of archives. After this operation, 2,296 kB of additional disk space will be used. Do you want to continue? [Y/n] Y (Reading database ... 176266 files and directories currently installed.) Removing jitsi-meet-tokens (1.0.962-1) ... dpkg: error processing package jitsi-meet-tokens (--remove):  subprocess installed post-removal script returned error exit status 10 Errors were encountered while processing:  jitsi-meet-tokens E: Sub-process /usr/bin/dpkg returned an error code (1)   Can someone help me figure out what I need to do to get this resolved? I'm fine with deleting jitsi, but I'm not sure if that will correct the problem.   Currently there is a red circle with a line through it in the upper right corner that indicates there may be installed packages with unmet dependencies. Prove that $(n+1)^{n-1}<n^n$ How would one prove that  $$(n+1)^{n-1}&lt;n^n \ \forall n&gt;1$$  I have tried several methods such as induction. Cite in the caption of a graphic but not in \listoffigures? Sometimes I'd like to cite the reference in the caption of a graphic. But I want to avoid that this citation is displayed in the \listoffigures. How can I achieve that?  Fore example with this code:  \begin{figure}[htb] \centering \includegraphics{overview_pyramid.png} \caption{Overview pyramid \cite{LM2010}}\label{fg:overview_pyramid} \end{figure} I know Americans commonly use &quot;damn&quot; or &quot;damned&quot; to describe things. Sometimes, more appropriately, it's even &quot;darn&quot; or &quot;darned&quot;. For example,  This damn/damned computer is too slow.  However, I can't wrap my head around when to use &quot;damn&quot; and when to use &quot;damned&quot;. Or is only one of them correct? I was pondering a bit more about  question regarding being able to &quot;omit&quot; the base case in a proof by strong induction due to vacuous truth. The post states:  Strong induction proves a sequence of statements $P(0), P(1), …$ by proving the implication &quot;If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true.&quot; for every nonnegative integer $n$. There is no need for a separate base case, because the $n=0$ instance of the implication is the base case, vacuously.  However, if we consider $n=0$, we would have that the statement is vacuously true, which I would take to mean that the implication is true regardless of the validity of $P(0)$. However, clearly it's necessary for $P(0)$ to hold for an induction proof to be valid. So I'm confused on how, by omitting the base case, $n=0$ isn't just a tautology, making the implication true regardless of whether $P(0)$ actually holds. How to make listings code indentation remain unchanged when copied from PDF? So I am using  package for code examples. But there is a problem. When I copy the code examples from the final PDF document, they are usually not the same - the indentation is gone, there are random spaces after parenthesis, etc. Does anyone know how to make sure that the code that I copy from the final document is the same as I wrote it in the LaTeX document?  Here is what I mean. Compile this document:  \documentclass[12pt,oneside]{memoir}  \usepackage{listings} \usepackage[T1]{fontenc} \usepackage{xcolor} \usepackage{textcomp}  \definecolor{codebg}{HTML}{EEEEEE} \definecolor{codeframe}{HTML}{CCCCCC}  \lstset{language=Awk} \lstset{backgroundcolor=\color{codebg}} \lstset{frame=single} \lstset{framesep=10pt} \lstset{rulecolor=\color{codeframe}} \lstset{upquote=true} \lstset{basicstyle=\ttfamily} \lstset{showstringspaces=false}  \begin{document}  This code example prints out all users on your system:  \begin{lstlisting}[language=c] #include &lt;stdio.h&gt; #include &lt;stdlib.h&gt; #include &lt;string.h&gt;  #define MAX_LINE_LEN 1024  int main() {     char line[MAX_LINE_LEN];     FILE *in = fopen("/etc/passwd", "r");     if (!in) exit(EXIT_FAILURE);      while (fgets(line, MAX_LINE_LEN, in) != NULL) {         char *sep = strchr(line, ':');         if (!sep) exit(EXIT_FAILURE);         *sep = '\0';         printf("%s\n", line);     }     fclose(in);     return EXIT_SUCCESS; } \end{lstlisting}  \end{document}   Now compile it, open it, select and copy the code example and paste it in a text editor. Here is the result I get:  # include &lt;stdio .h&gt; # include &lt;stdlib .h&gt; # include &lt;string .h&gt; # define MAX_LINE_LEN 1024 int main () { char line [ MAX_LINE_LEN ]; FILE *in = fopen ("/ etc / passwd ", "r"); if (! in) exit ( EXIT_FAILURE ); while ( fgets (line , MAX_LINE_LEN , in) != NULL ) { char * sep = strchr (line , ':'); if (! sep ) exit ( EXIT_FAILURE ); * sep = '\0 '; printf ("%s\n", line ); } fclose (in ); return EXIT_SUCCESS ; }   Jeez! What's this... The indentation is gone. There are spaces in included headers &lt;stdio .h&gt;, there are spaces after parenthesis [ MAX_LINE_LEN ].  Anyone knows how to fix this? I'd like the code examples in my book to be copy/paste-able into a text editor so people can try them out easily. Is the 64-Bit version of Ubuntu only compatible with AMD CPUs? I was told that computers with more than 2 gig's memory need a 64 bit operating system to utilize all RAM.  Is the 64bit Ubuntu download really JUST for AMD processors? I am asking because the disk image I downloaded says AMD64.  So will my new Intel 2.3Ghz Core i3 Dual Core processor work with 64 bit Ubuntu?  It runs the 64bit version of Windows without any qualm. One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus.  Now Schrödinger describes the electron as a wave function. His theory is able to describe all atoms (in contrast to Bohr's model), but how is the radiation problem solved? I understand that the wave has not an exact position in the time anymore. But the electron still "moves", so it has an acceleration anyway (because of vibrations or so).   Why is in this theory the electron not radiating anymore? And if it is, why doesn't the atom collapse? How to make 'python' program command execute Python 3? What is the lifetime of a static variable in a C++ function? How can I play Pokémon GO without Internet? What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography? Removing nodes from an XmlDocument The following code should find the appropriate project tag and remove it from the XmlDocument, however when I test it, it says:  The node to be removed is not a child of this node.  Does anyone know the proper way to do this?  public void DeleteProject (string projectName) {     string ccConfigPath = ConfigurationManager.AppSettings["ConfigPath"];      XmlDocument configDoc = new XmlDocument();      configDoc.Load(ccConfigPath);      XmlNodeList projectNodes = configDoc.GetElementsByTagName("project");      for (int i = 0; i &lt; projectNodes.Count; i++)     {         if (projectNodes[i].Attributes["name"] != null)         {             if (projectName == projectNodes[i].Attributes["name"].InnerText)             {                                                                 configDoc.RemoveChild(projectNodes[i]);                 configDoc.Save(ccConfigPath);             }         }     } }   UPDATE  Fixed. I did two things:  XmlNode project = configDoc.SelectSingleNode("//project[@name='" + projectName + "']");   Replaced the For loop with an XPath query, which wasn't for fixing it, just because it was a better approach.  The actual fix was:  project.ParentNode.RemoveChild(project);   Thanks Pat and Chuck for this suggestion. Reading a bit about recent Monica Cellio related posts, I am afraid that I may be using a language that is a lot like English, but might not be the Stack-Exchange-approved version of English. For example, my English has had a certain gender-neutral pronoun in living use for centuries. That pronoun is used throughout the literature written in this version of English. It is used in the English I was taught as a grammar school student and later as a graduate student working on a master's thesis. I believe that this English is good enough to communicate about topics relevant to the various Stack Exchange sites.  The last thing I want from Stack Exchange is to be booted out because I slip up and accidentally use the language that has served me so well for a lifetime.  The topics discussed on Stack Overflow (I almost referred to it as a "sister site;" would that have been allowed? "mother site?" Good thing I didn't?) are often centered on languages which have formal descriptions.  Where is the formal description of the language that is to be used on Stack Exchange?     Strive not with a man without cause, if he have done thee no harm.      He shall die without instruction; and in the greatness of his folly he shall go astray. I was wondering this today, and my algebra professor didn't know the answer.      Are subgroups of finitely generated groups also finitely generated?    I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?  And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?  NOTE:  This question has been merged with another question, asked by an undergraduate.  For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question. E: Unable to correct problems, you have held broken packages Trying to figure what language this PDA (pushdown automata) accepts I have the following PDA and I can't figure our what words it accepts, would like to get some help with figuring this out.    Of course it only accepts if that stack is empty in the accepting state. Preventing hyphenation of specific short, repeated, hyphenated words Generating hatched filling using Region functionality I am trying to boot Ubuntu on my computer.  When I boot Ubuntu, it boots to a black screen.  How can I fix this? Table of Contents: