How Prove this $ \sum\limits_{k=1}^{2n} { 2n \choose k } { 2n \choose 2n-k } = 2^{4n} - 1$
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
Error: class X is public should be declared in a file named X.java When you edit a message and repost the contents of  its first revision, the repost is eaten up by chat.  Steps to reproduce:   Join any chat room Post something. Let's say you post abcd Edit that to something else. For example, --- Try to post abcd again. The green (pending) message will show up, but when the response is received, it disappears (the server spits out {"id":null,"time":null} as the response)   This happens more often than you think. Many a time I post something, only to realize that I want to use the magic separator (---) to separate it from a previous block. So I up-arrow, Ctrl-X, and type ---, and then I repost the cut text. Which gets eaten up by the server -- making a small change to the repost fixes things. How do RAM and processor speed affect overall performance on Android? Since Android can be installed on wide range of devices, I'm wondering how important RAM and the CPU speed are to the performance of a device.  There are always tradeoffs when something is increased and another one decreased!  Is more RAM or a faster processor better for general browsing and less mulititasking?  What if I'm interested in gaming or doing lots of things at once?  Will more RAM make up for a slower processor or vice-versa? How do I get a Ralink RT3290 wireless card working? How to show or hide Keyboard Viewer with a keyboard shortcut? I would like to display and hide the Keyboard Viewer using a shortcut.   Is there a way to display the Keyboard Viewer via a keyboard shortcut? Is there a way to automatically display the Keyboard Viewer once I switch to a specific language? Physically Identify the failed hard drive Linked and Related sections contain duplicate links to questions The Linked and Related sections sometimes contain a link to same same question.  See for example , where one of the Linked questions is same as the first of the Related questions.  I think that this is redundant and duplicate links should be removed from Related section. Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in  real life most of the things we do are not linear functions and derivatives helps to make a real life functions into linear.  eg converting parabola into a linear function $x^{2}\rightarrow 2x$  but then i find this; derivation of $\sin{x}\rightarrow \cos{x}$  why we cant use $\sin{x}$ itself to solve the equation. whats the purpose of using its derivative $\cos{x}$. Please forgive me if i have asked a stupid questions, i want to improve my fundamentals in calculus. Please anyone tell me when we have to use "has been, have been , had been" and something  started in past and still continuing is it possible to use has been ? Tangent lines to ellipse from external point I have a set of 10 tables and each has an extra column that I don't want. Specifically, these tables are descriptions of database tables exported by phpMyAdmin, and they all have a "MIME" column that I would like to remove. Is there a way to do this other than using string manipulation / find and replace?   I have tried setting the column to p{0in}, but the column title still remains. What are the best strategies for assessing if a journal is a &quot;vanity&quot; or &quot;predatory&quot; journal that should be avoided (both for publishing in and reviewing for)? For example, how would one go about determining if a journal/publisher belongs on ? Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$ Apostol's Calculus, exercise number I 4.7 13.     Prove that if $n \geq 1$, then   $$ 2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1}) $$   and use this to prove that if $m \geq 2$, then   $$ 2 \sqrt{m} - 2 &lt; \sum_{n = 1}^m \frac{1}{\sqrt{n}} &lt; 2\sqrt{m} - 1 $$   In particular, when $m = 10^6$ the sum lies between $1998$ and $1999$.   After establishing the trivial case, for $n=1$, I can't think of a way to perform the inductive step:  assuming $2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1})$ and using this, deducing the same for $n+1$ is true as well:  $2(\sqrt{n+2} - \sqrt{n+1}) &lt; \frac{1}{\sqrt{n+1}} &lt; 2(\sqrt{n+1} - \sqrt{n})$  Any help is very appreciated, thanks How can I disable landscape mode in Android? I am a beginning indie developer, and I want to get the word out about my game. What sites can I go to to ask about reviews of my game, and where might I be able to talk about and get feedback on my game from players? Evaluate $\int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos(t)^3\sin (t)^3)dt$ Call dynamic_sidebar but include/exclude named widgets? Is it possible to include or exclude specific named widgets that are assigned to a named dynamic_sidebar call?  For example, if I've registered a sidebar named "my_sidebar" and the user had placed a "Links" widget into it, I want to be able to include or exclude it based on a custom setting in my theme options panel.  Is this possible?  Any insights much appreciated. I was trying to install Teamviewer, but I was getting a dependency error. I tried to install the required packages, but with no luck.  I'm getting this error:  Unpacking teamviewer (from teamviewer_linux_x64.deb) ... dpkg: dependency problems prevent configuration of teamviewer:  teamviewer depends on lib32asound2; however:   Package lib32asound2 is not installed.  teamviewer depends on lib32z1; however:   Package lib32z1 is not installed.  teamviewer depends on ia32-libs; however:   Package ia32-libs is not installed.  dpkg: error processing teamviewer (--install):  dependency problems - leaving unconfigured Errors were encountered while processing:  teamviewer   I tried   sudo apt-get -f install   Getting   Package ia32-libs is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source However the following packages replace it:   lib32z1 lib32ncurses5 lib32bz2-1.0  Package lib32asound2 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'lib32asound2' has no installation candidate E: Package 'ia32-libs' has no installation candidate   I can't even get to the   sudo dpkg -i teamviewer_linux_x64.deb   If I force installation  sudo dpkg --force-depends -i teamviewer_linux_x64.deb   Although it's "Setting up Temviewer" it gives me this:    How can I fix this problem? What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)? I've read a lot about PCA, including various tutorials and questions (such as , , , and ).   The geometric problem that PCA is trying to optimize is clear to me: PCA tries to find the first principal component by minimizing the reconstruction (projection) error, which simultaneously maximizes the variance of the projected data.    When I first read that, I immediately thought of something like linear regression; maybe you can solve it using gradient descent if needed.  However, then my mind was blown when I read that the optimization problem is solved by using linear algebra and finding eigenvectors and eigenvalues. I simply do not understand how this use of linear algebra comes into play.  So my question is: How can PCA turn from a geometric optimization problem to a linear algebra problem? Can someone provide an intuitive explanation?  I am not looking for an answer like  that says "When you solve the mathematical problem of PCA, it ends up being equivalent to finding the eigenvalues and eigenvectors of the covariance matrix." Please explain why eigenvectors come out to be the principal components and why the eigenvalues come out to be variance of the data projected onto them  I am a software engineer and not a mathematician, by the way.  Note: the figure above was taken and modified from .