$$\sum_{0 \le k \le a}{a \choose k}{b \choose k} = {a+b \choose a}$$  Is there any way to prove it directly?  Using that $\displaystyle{a \choose k}=\frac{a!}{k!(a-k)!}$?
How can we prove that     $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$   (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
Can I install Android on my non-Android device? Many people want to try/use Android without buying a new Android phone.  How can I install Android on my current phone or device running Nokia Symbian, Maemo, or MeeGo; Microsoft Windows Mobile or Windows Phone 7/8/10; Apple iOS (iPhone); RIM's BlackBerry OS; Palm's WebOS; Samsung's Bada; HP's WebOS; Mozilla's Firefox OS; or other phone / operating system? What sites should have hyperlinking (onebox) support in chat? Some sites are . What additional sites should the chat support?  One suggestion per answer. How do I calculate the date six months from the current date using the datetime Python module? Tool to create text files from a template I have to regularly create 100+ text files from templates. I currently use an overly complicated shell script. I think there is a smarter way to handle this, but I don't know how.  I have a "database" :  # outputfile      template            data1   data2    data3 first.txt         $template_main      $text1  abcd     1234 second.txt        $template_main      $text2  efgh     5678 third.txt         $template_other     $text1  ij       90   And a configuration file :  template_main=main.txt template_other=other.txt text1=whatever text2=blah   The templates are text files with placeholders like %%data2%% (the placeholder form can be changed).  Does someone know a tool to automate this better than with a complicated shell script ? Why does the (relativistic) mass of an object increase when its speed approaches that of light? I'm reading Nano: The Essentials by T. Pradeep and I came upon this statement in the section explaining the basics of scanning electron microscopy.     However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. At such velocities, one needs to do relativistic correction to the mass so that it becomes[...]   We all know about the famous theory of relativity, but I couldn't quite grasp the "why" of its concepts yet. This might shed new light on what I already know about time slowing down for me if I move faster.  Why does the (relativistic) mass of an object increase when its speed approaches that of light? Trouble with proving $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain can't find using System.Web.UI.HtmlControls namespace please I tried to find counterexamples to see that $l^p$ is not norm with $p&lt;1$ in the triangle inequality but I have problems with convergence when I choose some successions. Thanks. How can I specify a long list of math operators? The preambles of my LaTeX documents often include many many lines of \DeclareMathOperator instructions, e.g.  \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Tet}{Tet} \DeclareMathOperator{\Maps}{Maps} \DeclareMathOperator{\Diff}{Diff}   Is there a nice way to map some macro over a list, so I could do this more concisely, and with less copying and pasting when I add a new math operator? In Batman #1 in the New 52, when we visit the Batcave for the first time, we see a dinosaur:     What... is up with that? It is a completely awesome cave decoration - but what else is it? Is it a non sequitur or is it a call back to an earlier storyline? I imagine it's the latter – a life-size T-rex model is an expensive piece of decor, even for Bruce Wayne.  Bonus points: are there any other relics from past storylines visible in this Batcave?  Update: here's a shot of the other panel that gives us a view of the "showroom" (including the huge dice). Regex to validate date format dd/mm/yyyy For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$ Here are my thoughts so far:    $C_p = \langle x \mid x^p =e\rangle$ $C_q = \langle y \mid y^q =e\rangle$ $C_p \times C_q$ has elements of the form $(x^a,y^b)$ There are $p$ possible values for $x^a$ and $q$ possible values for $y^b$. So there are $pq$ possible elements in $C_p \times C_q$. $C_{pq} = \langle z\mid z^{pq} = e\rangle$ and there are $pq$ elements in this group. For an isomorphism from $C_p \times C_q$ to $C_{pq}$ we send $(e,e)$ to $e$ to ensure that the identities are mapped to each other. However, I'm not sure how to define an isomorphism for the other elements. Can I install the 64 bit version? I am a permanent resident (U.S) and I'm crossing the border into Mexico and taking a flight to Mexico city. The passport from my home nation is expired and I'm trying to avoid the last minute 500 mile+ trip to the nearest embassy. My question is, can I get on the plane with my green card or state license? or my expired passport and birth certificate? also, can i get the FMM/tourist card with my drivers license? all the official information says i need a valid passport, but other travelers have had different experiences, does it just depend on the officer? I already purchased my flight! In Java, I'm dynamically creating a set of files and I'd like to change the file permissions on these files on a linux/unix file system.  I'd like to be able to execute the Java equivalent of chmod. Is that possible Java 5? If so, how?  I know in Java 6 the File object has setReadable()/setWritable() methods. I also know I could make a system call to do this, but I'd like to avoid that if possible. Recently my paper is reviewed (by a professor from independent third party) and accepted by a journal. But, my employer allows me to publish this paper only without its name there.  Publishing house is expecting affiliation for each author, i.e. I can not leave affiliation as blank.   What should I write as the affiliation? Root drive is running out of disk space. How can I free up space? I created a simple service layer polygon in network analyst (ArcGIS Desktop 10.2) with no hierarchies set in the street network. Now I don't understand why is this polygon created along some particular streets?   Why isn't is properly spread out? For example, why doesn't the polygon cover the areas/streets I have marked in red? Obviously those marked areas are also within 1 km.  The impedance is 1 km and 2 km respectively.   The streets are also well connected individual line elements as you can see the junctions. How do you prevent web pages from appearing in the Google search index? How do I color intersection of two ellipses in different color. For instances, the ellipses should themselves be colored as yellow and their intersection as red. My code for drawing ellipses is given below  \draw (-4,0) ellipse (8 and 3); \draw (8,0) ellipse (8 and 3);   UPDATE: In the comments to individual answers, I have requested to show it for three ellipses.