This is a formula about the binomial.I am so confused to prove it by combination way.  $${\sum _{r=0}^{k}{\binom {n+r-1}{r}}={\binom {n+k}{k}}}$$
After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
When Deadpool is in the hallway talking      he makes a joke about Fox springing for a few more X-Men and the camera pans to show Beast closing the door to get rid of the noise.   Who else was in the room? Characterization convex function. Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help. What's the difference between :: (double colon) and -> (arrow) in PHP? What's the difference between denizen, resident, inhabitant? What's the difference between , , and ?  Even if their ODO definitions suggest they are interchangeable most of the time, are there fine differences in meaning between them? How do I install the latest OpenOffice? remove isp-provided dns My isp provides a dns service; but unfortunately they do block certain sites. I would like to use opendns or google; but it seems like it doesn't work.    I have set up a wired connection; given a manual ip and entered opendns-ip's in the nameserver field (all through the gui). Still url's are being resolved through my ISP. I have tried to configure the same on my girlfriends laptop (win 7) and I also can't seem to do the setup. My previous computer also was a win 7 and had no problem. (so I believe the problem is 'me') If it were only one site, i could edit the hosts file; but I prefere to circumvent their whole dns server.  Is there something an isp can do to block the use of other dns-servers than theirs? Is it possible to re-route the ISP dns to the opendns? Could I use an IPS based TV to edit photos in a color-calibrated environment? Over the past weeks, I've been trying to get all of my color issues worked out. Getting closer, but I'm not there. I've calibrated my monitor with a friend's calibrator and that helped. I'm using the profiles provided by the local processor.  It's close. How close depends on the viewing angle.  So I'm thinking IPS monitor might be a good next step.  Googling around, I see that I could get an   Given that IPS based 22/23" monitors run $250-$300, this seems like a lot more monitor for not much more money.  What are the faults in this logic? Is there an easier or more 'correct' way to open gedit as root? I have ubuntu 12.04 running in a virtual machine on virtual box.  Its existence is mainly as a 'toy' to play about with Ubuntu, so for this reason I wanted the GUI (is that what 'Dash' is in ubuntu 12.04?  Anyway...)  One thing I am currently using this machine for is a LAMP server.     I use 'gedit' to edit my files, but because I haven't correctly set permissions yet I just open it with sudo in terminal.  Just now, to try to simplify this process I created a shell script on the desktop with a very simple job - sudo gedit  I can't help thinking there must be an even more obvious way to open gedit as root from the GUI than this.  So is there?  Edit:  I just 'improved' my script by replacing sudo with gksudo.  Now I don't have to have a terminal window open while gedit is open.   So I guess this is quite convenient for now :)   It would be good, for the sake of knowledge, to know if this is anything close to how the more seasoned ubuntu users would do this type of thing. Problem with defining shortcuts for TikZ matrices For  I wanted to define a shortcut for creating a matrix inside a tikzpicture. As a minimal example, consider  \newcommand\mymatrix[1]{     \begin{tikzpicture}         \matrix[matrix of math nodes] {#1};     \end{tikzpicture}% }   which would then be used as  \mymatrix{     a &amp; b \\     e &amp; f \\ }   But LaTeX complains about Package pgfbasematrix Error: Single ampersand used with wrong catcode at the line with the closing brace of \amatrix{}. Why does it do that and how can I avoid it? What is god for religious people? In order to discuss with religious friends about god, I would like to know if there's some accepted objective definitions of God for main religions, even if belief is subjective.   For example, is he "a person who..."? In that case I would address features of a person, as capabilities and limitations. Is he "an entity..."? In that case I would use systems theory.   I feel lost on this, being agnostic from a religious family, where I was taught that God is a mess of things, including a beautiful bearded face, omnipotence and a mix of human defects and metaphysical powers. I'm hoping someone can give me a nudge in the right direction...     Let $F$ be a finite field, and let $f(x)$ be a nonconstant polynomial whose derivative is the zero polynomial.  Prove that $f$ cannot be irreducible over $F$.   I've got that every root of $f$ is a multiple root and that for $F=\mathbb{F}_{p^r}$, the exponent of every term of $f$ is a multiple of $p$. If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is  $\bf{My\; Try::}$ Given $11z^{10}+10iz^9+10iz-11 = 0\Rightarrow \displaystyle z^9 = \frac{11-10iz}{11z+10i}.$  Now Put $z = x+iy\;,$ we get $\displaystyle (x+iy)^9 = \frac{11-10i(x+iy)}{11(x+iy)+10i} = \frac{(11+10y)-10ix}{11x+i(10+11y)}$  Now i did not understand how can i solve it,  Help Required  Thanks Some of the Puella Magi seem to have magical powers related to the wishes that they made. Sayaka has regeneration abilities, and she wished to heal someone. Homura has time manipulation abilities, and she wished to go back in time.  But other magical girls, like Mami and Kyouko, don't appear to have powers related to their respective wishes.  So, is there a connection, or is it only a coincidence for Sayaka and Homura? If it's the former, how do Mami and Kouko's wishes relate to their powers? In my team, we use Selenium for automation of functional testing. We also do a lot of manual regression testing to make sure the appearance of the rendered web pages is ok. I would like to develop a framework, that would allow comparing screenshots of the "base" live web page, and the web page on the test server after applying the software update. I believe this will drastically reduce the amount of manual regression testing we need to do.   I would like the framework to be able to detect portions of web pages that were rendered differently from each other. An example of such functionality can be  (first image in the article)  I have never worked with image processing and would need ideas on what algorithms are already publicly available that would help me with the task. From what I can see, the simple pixel-by-pixel comparison would not work well, because it does not account for the concept of elements.  Or am I approaching this problem from a completely wrong direction? How to increase the width and height of the Rich Text area field in Visual force Page? Is hyperref really the best way to add metadata to a TeX file? So I know one of hyperref's features is that it can add metadata to you PDFs. This is a nice feature and if you're putting stuff online, it's good to include metadata.  But is the hyperref package really the best way to do it? I mean, if you're using hyperref anyway, that's fine. But if you don't want hyperrefs in your document, it seems weird to include the package, and then turn off pretty much all its functionality. And given that hyperref is a shorthand for "package conflict nightmare", is seems insane that you should have to deal with it just for the sake of some metadata.  So I have two questions.   Is there a good reason that hyperref includes this functionality which is apparently unrelated to its primary purpose? Is there a better (smaller, less conflicty, easier...) option to include metadata in my PDFs? Downloading all my questions with the answerer names I am writing several papers in which I use answers which I received in various StackExchange websites. Naturally I would like to acknowlege and thank the SE members that gave the useful answers. But there are many such answers, and each answer is relevant to a different paper.   So, I would like to download all the questions that I asked in the various SE websites, including the answerer names, to an Excel worksheet in which I can organize the questions by subject.  Is this possible? I made a huge project, wanted to reduce file size from 1.3GB to below 1GB. Compressing did not help. I removed objects from scene. Also textures. I then realized, the file size even was the same when I delete ALL objects. Why?  Download of sample:   Before deleting all: 1.2GB, after: 1.1GB. Schengen visa validity start date My Chinese in-laws intend to visit us in one month from now. Their visa is valid from Nov 23rd. Would it be possible for them to take the airplane on November 22nd in China and arrive in Schengen area on November 23rd?  Or does the visa already have to be valid upon departure?  In our particular case, they still need to transfer in Beijing first. The actual flight to Schengen zone effectively only departs on the 23rd so I guess that should be completely fine then? How to distribute proprietary Mathematica code