Four cards are drawn from a pack of 52 cards. What is the chance that no two cards are of equal value? Each belongs to a different suit? My try-- Total probability =$^{52}C_4$ In the pack of 52 cards there are 4 aces, 4 1s and so on. So, there will be 13 groups of 4 cards containing same value. $\therefore$ Probability that no two cards are of equal value =$\frac{^{13}C_4\times ^4C_1}{^{52}C_4}=\frac{44}{4165}$ But given answer is=$\frac{2816}{4165}$ I think I did it wrong, please help me. I also couldn't solve the 2nd problem.
4 cards are drawn from a pack without replacement. What is the probability of getting all 4 from different suits?  Here's how I tried to solve:  For the first draw, we have 52 cards, and we have to pick one suit. So, probability for this is $\frac{13}{52}$.  For the second draw, only 51 cards are left. The second suit has to be selected, so there are 13 cards from that suit. The probability is $\frac{13}{51}$.  Similarly, the third and fourth draw have probabilities $\frac{13}{50}$ and $\frac{13}{49}$ respectively.  Since the draws are independent, the total probability becomes $$\frac{13}{52} \times \frac{13}{51} \times \frac{13}{50} \times \frac{13}{49}$$  But my book says the answer is $\frac{{13\choose 1} \times {13 \choose 1} \times {13\choose1} \times {13\choose1}}{52 \choose 4}$.  My answer differs by a factor of $4!$. What did I do wrong?
Long-distance movement arrow in gloss I have a question on drawing an arrow using gb4e. Currently, I am trying to use the method described in the link below. The problem takes place when I draw the long distance arrow like (5). As you can see the picture below, the line cross over the gloss. I would like to draw the horizontal line like the example (1). How can I evacuate the crossing?    \documentclass[11pt, dvipdfmx]{article} \usepackage{tikz} \usetikzlibrary{calc} \newcommand\tikzmark[1]{\tikz[remember picture, baseline=(#1.base)] \node[anchor=base,inner sep=0pt, outer sep=0pt] (#1) {#1};}  % This code from http://tex.stackexchange.com/q/55068/2693 \tikzset{     ncbar angle/.initial=90,     ncbar/.style={         to path=(\tikztostart)         -- ($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)         -- ($(\tikztotarget)!($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztostart)$)         -- (\tikztotarget)     },     ncbar/.default=0.5cm, }  % Thanks to Paul Gessler adn Percusse for code improvement here \newcommand{\arrow}[2]{\begin{tikzpicture}[remember picture,overlay] \draw[-&gt;,shorten &gt;=3pt,shorten &lt;=3pt] (#1.base) to [ncbar=\arrowht] (#2.base); \end{tikzpicture} \setlength{\arrowht}{0ex} } \usepackage{gb4e} % The following code modified from  % http://permalink.gmane.org/gmane.comp.tex.linguistics/1036 % This adds some extra space after the first line \newlength{\arrowht} \setlength{\arrowht}{0ex} \newcommand*\cgdepthstrut{{\vrule height 0pt depth \arrowht width 0pt}} \renewcommand\eachwordone{\cgdepthstrut\rmfamily} \renewcommand\glt{\vskip -\topsep} \let\trans=\glt \newcommand\arrowex{\setlength{\arrowht}{2.5ex}\ex}   \begin{document} \begin{exe} \ex\label{scramble-det} \begin{xlist}         \arrowex          \gll          \ldots dat Jan \tikzmark{een} auto gisteren \tikzmark{t}  gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday.         \arrow{t}{een} \end{xlist} \end{exe}  \begin{exe} \ex\label{scramble-dets} \begin{xlist}         \ex          \gll          \ldots dat Jan een auto gisteren t gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday. \end{xlist} \end{exe}  \begin{exe} \ex[]{     \gll     John-wa totetumonaiukurai baka-da.    \\     John-Top very stupid-cop \\     `John is very studip.'     }  \ex[]{     \gll     John-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very studip.'     }  \arrowex[]{     \gll     \tikzmark{John}-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai  \tikzmark{t} baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very stupid.'         \arrow{t}{John} }  \end{exe}  \end{document} I'm using the gnome flashback session with metacity window manager, but I can't change to any metacity theme in any of the tweaking tools, as they're not even appearing in the lists. Is there anything I can do? How to avoid the pop up message while downloading an application (.exe) using selenium I am trying to download an application and check if it's downloaded into my local system or not. But when I click the download link I have a popup with Save or Cancel options, which is stopping me from downloading the file. I googled and found some solutions and tried to change the profile of Firefox by adding the below code, but it is not working.  firefoxProfile.setPreference("browser.download.folderList", 2); firefoxProfile.setPreference("browser.download.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.helperApps.neverAsksaveToDisk", "application/x-msexcel,application/excel,application/x-excel,application/excel,application/x-excel,application/excel,application/vnd.ms-excel,application/x-excel,application/x-msexcel"); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("browser.download.manager.focusWhenStarting", false); firefoxProfile.setPreference("browser.helperApps.alwaysAsk.force", false); firefoxProfile.setPreference("browser.download.manager.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.download.manager.closeWhenDone", false); firefoxProfile.setPreference("browser.download.manager.showAlertOnComplete", false); firefoxProfile.setPreference("browser.download.manager.useWindow", false); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("services.sync.prefs.sync.browser.download.manager.showWhenStarting", false);   This is the code I added for setting up Firefox preferences, but I still get the popup. Is there a "smallest" divergent sum? A function $f : I \rightarrow \mathbb R$ on an interval $I$ is convex   if $f((1-t)x+ty)\le (1-t)f(x)+tf(y), \forall x,y \in I, t \in [0,1]$      Assume now that $I$ is an open interval.  Show that if $f$ is convex   then for each $c \in I$ there exists $m \in \mathbb R$ such that $m(x  − c) + f(c) \le f(x)$ for all $x \in I$ , and if in addition $f$ is   differentiable at $c$ then $f'(c)$ is the unique $m$ that works. In   general, must m be unique?   (we have previously shown that convex functions are continuous)  I think I'm missing the point of the question.  Surely, by the mean value theorem we can find some $y\in (x,c)$ such that $f'(y)=\frac{f(x)-f(c)}{x-c}$ setting $m=f'(y)$ we get $m=\frac{f(x)-f(c)}{x-c}, m(x-c)+f(c)=f(x)$ such an $m$ fits the inequality we're asked to show.  I also cannot see why $f'(c)$ would be the unique solution or why we would ever get a unique solution of the inequality at all for that matter.  Clearly I'm misunderstanding the question, so if someone could point out where I would appreciate that.  Thank you Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$. Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.   where $\sigma (n)$ is the sum of all the divisors of $n$ and $\sum\nolimits_{d|n} f(d)$ is the summation of $f$ at each $d$ where $d$ is the divisor of $n$.    I have written $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}.......p_k^{\alpha_k}$ then:- $$\begin {align*} \sum\nolimits_{d|n} \frac{1}{d}&amp;=\frac{d_2.d_3......d_m+d_1.d_3......d_m+........+d_1.d_2.d_3......d_{m-1}}{d_1.d_2.d_3......d_m}  \\&amp;=\frac{d_2.d_3......d_m+d_1.d_3......d_m+........+d_1.d_2.d_3......d_{m-1}}{p_1^{1+2+...+\alpha_1}p_2^{1+2+...+\alpha_2}p_3^{1+2+....+\alpha_3}.......p_k^{1+2+....+\alpha_k}}  \\ \end{align*}$$ where $d_i$ is some divisor among the $m$ divisors. Then I cannot comprehend the numerator so that to get the desired result.  Also suggest some other approches to this question. If the topology of a space  has a countable base, then each base contains a countable subfamily which is also a base. Russian babel and chemformula Why do some people use $+\infty$ instead of $\infty$? Why do some people use $+\infty$ instead of $\infty$? Because we usually denote minus infinity as $-\infty$, I think it is sufficient to denote plus infinity as $\infty$. Are there any reasons for this notation? I saw that many people majoring analysis use like $\int_0^{+\infty}$ or $\sum_{j=1}^{+\infty}$ or $\bigcup_{j=1}^{+\infty}$ which is not related to the unsigned infinity. Git for beginners: The definitive practical guide So that I could resize the window to a certain size from within Terminal. I know there is a way to view all my questions in reverse chronological order from the PC by going to my profile. But I would like to do the same from the mobile app for Android. On Android in the left upper hand corner I can only see the latest answers to my posts, but I would be able to view all my posts. How to upgrade to MySQL 8.0? I have a Drupal site 8.6.10 on an Ubuntu server 18.04 with MySQL 5.7.25  I want to update MySQL to version 8.0  Here are the steps I followed :   I backed up my database. I uninstalled MySQL from my server with the commands :  $ sudo systemctl stop mysql $ sudo apt remove mysql-* $ sudo apt purge mysql-* $ sudo apt autoremove $ sudo dpkg -l | grep mysql | grep ii  I added the repository with the command :  $ wget https://repo.mysql.com//mysql-apt-config_0.8.12-1_all.deb  I installed the package with the command :  $ sudo dpkg -i mysql-apt-config_0.8.12-1_all.deb            My problem :   I install MySQL with the command :  $ sudo apt update $ sudo apt install mysql-server    When I do step 5, it installs MySQL 5.7 why does not it install version 8 ? Transvection matrices generate $ \operatorname{SL}_n(\mathbb{R}) $ Create instance of generic type in Java? I've been exploring implementing Angular 2.0 in Salesforce custom pages and found that if I set showHeader=true in the VF page, I run into some issues.   It looks like including the header loads "sfdc/main.js", which defines the browser's Map() function without checking if one already exists. This removes some needed prototype functions that are native to ES5/6 such as "set" (needed in Angular 2) and "delete" (needed for an ES6-shim).   I've found that even including an ES5 shim before the ES6 one in attempts to restore some of these prototype methods does not work. If I set showHeader=false, "sfdc/main.js" is not loaded and everything works fine.   Is there a recommended way around this, or is it possible to update "sfdc/main.js" to conditionally define Map() if one does not already exist? It looks like it leaves out quite a few native functions supported in modern browsers.  Thanks I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches.  Recently I've bought Topology by D. Kahn and am finding it difficult.  Here is a problem that I think I'm am answering sufficiently but any help would be great if I am off.  If $d$ is a metric on a set $S$, show that  $$d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is a metric on $S$.  The conditions for being a metric are $d(X,Y)\ge{0}, d(X,Y)=0$ iff $X=Y$, $d(X,Y)=d(Y,X)$, and $d(X,Y)\le{d(X,Z)+d(Z,Y)}$.  Thus, we simply go axiom by axiom.  1)  Since both $d(x,y)\ge{0}$ and $1+d(x,y)\ge{0},$ it is clear that $d_1(x,y)\ge{0}$.  (Is this a sufficient analysis?) 2) $d_1(x,x)=\frac{d(x,x)}{1+d(x,x)}=\frac{0}{1+0}=0$. 3) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}=\frac{d(y,x)}{1+d(y,x)}=d_1(y,x).$ 4) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}\le{\frac{d(x,z)+d(z,y)}{1+d(x,z)+d(z,y)}}=\frac{d(x,z)}{1+d(x,z)+d(z,y)}+\frac{d(z,y)}{1+d(x,z)+d(z,y)}\lt\frac{d(x,z)}{1+d(x,z)}+\frac{d(z,y)}{1+d(z,y)}=d_1(x,z)+d_1(z,y).$  However, #4 is strictly less, not less than or equal to, according to my analysis, so where did I go wrong? What level are spells cast at if a warlock multiclasses as another caster? How to differ sessions in browser-tabs? Computational complexity of Fibonacci Sequence I understand Big-O notation, but I don't know how to calculate it for many functions. In particular, I've been trying to figure out the computational complexity of the naive version of the Fibonacci sequence:  int Fibonacci(int n) {     if (n &lt;= 1)         return n;     else         return Fibonacci(n - 1) + Fibonacci(n - 2); }   What is the computational complexity of the Fibonacci sequence and how is it calculated?