I'm trying to prove this combinatorially. $$k\binom{n}{k} = n\binom{n-1}{k-1}$$ I know the first step is to relate a question to the equation. My question was if you have $n$ friends how many ways can you choose $k$ of them. I know this isn't correct because that would be the question if the left side wasn't being multiplied by k. Can anyone help me figure out what multiplying ${n \choose k}$ by $k$ means?
I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above?
Ok, so I have an index.php file which has to process many different file types.  how do I guess the filetype based on the REQUEST_URI.  If I request http://site/image.jpg, and all requests redirect through index.php, which looks like this  &lt;?php    include('/www/site'.$_SERVER['REQUEST_URI']); ?&gt;   How would I make that work correctly?  Should I test based on the extension of the file requested, or is there a way to get the filetype? I have an Ubuntu 12.04 machine on which I have installed VirtualBox with Ubuntu 12.04 as the guest OS.  How can I share files between the host and guest?   I have installed the guest additions also but still it’s not able to mount the sf_shared folders if even it’s mounted. This means I'm unable to retrieve the file from the host. How to print line numbers to the log. Say when outputting some information to the log, I also want to print the line number where that output is in the source code. As we can see in the stack trace, it displays the line number where the exception has occurred. Stack trace is available on the exception object.  Other alternative could be like manually including the line number when printing to the log. Is there any other way? Closing Reject Suggested Edit dialogue freezes review process If you go to review a suggested edit and close the Reject dialogue box, the button at the top-right will continue to say "Rejecting...". You will not be able to continue with the review process.  I have refreshed the page, and tried clearing my cache. If I refresh the page, I can continue reviewing the edit.  I'm using Chrome Version 31.0.1650.57 m. What diagnostic plots (and perhaps formal tests) do you find most informative for regressions where the outcome is a count variable?  I'm especially interested in Poisson and negative binomial models, as well as zero-inflated and hurdle counterparts of each. Most of the sources I've found simply plot the residuals vs. fitted values without discussion of what these plots "should" look like.  Wisdom and references greatly appreciated. The back story on why I'm asking this, if it's relevant, is .  Related discussions: How can I change iTunes country of region when having some credit in my account? I moved to Canada and I want to change my country of region from USA to Canada, but when I try to change it ( by click on my account in the iTunes Store and going to my Account Page) it showed me this error: "You have a store credit balance; you must spend your balance before you can change stores"    I must mention I have redeem 100$ to my account but I don't have any intention to spent it now, what is my option? Uninstalling Homebrew messed up git I uninstalled Homebrew using .  After running the above commands, I can't find .bashrc and .bash_profle. All files are gone from Documents , Desktop and Downloads :'(  When I do git status on /Users/ava it shows:  # On branch master # Untracked files: #   (use "git add &lt;file&gt;..." to include in what will be committed)  ... all the files which are in /Users/ava   history   113  cd `brew --prefix`   114  rm -rf Cellar   115  brew prune   116  rm `git ls-files`   117  rm -r Library/Homebrew Library/Aliases Library/Formula Library/Contributions   118  rm -rf .git   119  rm -rf ~/Library/Caches/Homebrew   120  sudo chown -R $USER:admin /usr/local   121  ruby -e "$(curl -fsSL https://raw.github.com/mxcl/homebrew/go)"   122  brew doctor   What just happened? What is the recommended way to encrypt user passwords in a database? Gradient descent and many other methods are useful for finding local minima in cost functions. They can be efficient when the cost function can be evaluated quickly at each point, whether numerically or analytically.    I have what appears to me to be an unusual situation. Each evaluation of my cost function is expensive. I am attempting to find a set of parameters that minimize a 3D surface against ground truth surfaces. Whenever I change a parameter, I need to run the algorithm against the entire sample cohort to measure its effect.  In order to calculate a gradient, I need to change all 15 parameters independently, meaning I have to regenerate all the surfaces and compare against the sample cohort way too many times per gradient, and definitely way too many times over the course of optimization.  I have developed a method to circumvent this problem and am currently evaluating it, but I am surprised that I have not found much in the literature regarding expensive cost function evaluations.  This makes me wonder if I am making the problem harder than it is and that there might be a better way already available.  So my questions are basically this: Does anyone know of methods for optimizing cost functions, convex or not, when evaluation is slow?  Or, am I doing something silly in the first place by rerunning the algorithm and comparing against the sample cohort so many times? Should I be offered the option to upgrade to 18.04.1? Does this double sum $\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{m+n+mn}{2^m(2^m+2^n)}$ converge? Editing photo dimensions in large batches If I had the time, I would go through each and every photo to make sure that the dimensions and quality are pixel perfect. However, I'm in a time crunch, and need to make the couple hundred photos that are 4320 x 3240 into 950x630.   now I could- 1. Upload them into WordPress, uses native image reducing components and J query go from there. 2. Go through each and every photo, reduce file size, and proportion and bring it to Yahoo's PNG processing script.  However, maybe I'm missing a choice or alternative workflow that would allow for processing all of these photos in a clear manner without losing quality or perspectives. Any workflow suggestions? Invest money and get 120% back - a site says How to find original community description Before joining a community you can easily see the community description. For example, for example before joining the Programming Puzzles &amp; Code Golf Stack Exchange you see the description:     Programming Puzzles &amp; Code Golf Stack Exchange is a question and   answer site for programming puzzle enthusiasts and code golfers.   However, after joining I can no longer see this description. I can go to the on-topic page and see a different description:     Programming Puzzles &amp; Code Golf is for programming contests and   challenges. We welcome questions from beginners and experts alike.   Yet, I cannot seem to find the original description anywhere. Is there a way to find the original description while being logged into a Stack Exchange site?  Based on Rene's answer, I can click the Tour link on each site, but StackOverflow (and maybe other sites) do not have a tour link. Instead, I'd have to go there by manually typing it in the address bar. Is there a reason that this has been removed from the Stack Overflow page? Can you nest html forms? Transit visa requirements at Narita Airport Since I am a regular Stack Overflow follower, I came across this and was shocked a bit regarding the design and user experience perspective.  Attaching picture 1 - it states that I can click any site and view a synopsis about it. Clear!!!    Attaching Picture 2 - you can see that when I clicked Webmasters, the design wants me to scroll horizontally to view all information and basically it hides information.    It would be nice, if it did something like below : Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$ While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? How do I run a batch file from my Java Application? Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$ Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$   Here's what I did so far:  Let $\sup A, \sup B$, and $\sup (A \cup B)$ denote upper bounds of $A,B$, and $A \cup B$, respectively. If $x \in A$, then $x \le \sup A$. If $x \in B$, then $x \le \sup B$. In either case, $x \in (A \cup B)$. So either $x \le \sup A$ or $x \le \sup B$. In either case, $x \le \max \{\sup A, \sup B\}$. So $\max \{\sup A, \sup B\}$ is the upper bound of $A \cup B$, which means $\sup (A \cup B) \le \max\{\sup A, \sup B\}$.  I do not know if there's a better way to go at it, but all I did was show the inequality. Can I do this, and show the reverse inequality? If so, how can I go about doing that?