$$\dbinom{2n}{n}=\sum_{k=1}^{n}\dbinom{n}{k}^2$$  Attempt: RHS gives the number of ways to choose some committee and then choose another committee such that some people may be in two of the committees.  The LHS shows the scenario when n people are picked from a group of 2n. I am having trouble showing how the LHS shows the same idea as the RHS.
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$  I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$  But I don't think it can lead me anywhere. I would like the proof to be as simple as possible.
Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects? Why are vias placed this way on a PCB? I used to check complex commercial PCBs specially those of graphics cards to see how professional PCB designers do their layout and learn from their techniques.  When I checked the card shown below I noticed two things regarding the placement of vias:  (A higher resolution image is ).   The PCB is surrounded by stitching vias all around the edges. What's the role of all of these? I think they're connected to ground to act as a shield, if that's true, I can't understand technically how by this placement they achieve this shield? By looking closer to the mounting holes, I noticed they added vias all around the pad, why? For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. I'm just trying to review basic terminal commands. Having said that, how do I create a text file using the terminal only? Why was my flag declined? I've recently flagged the following , which I got in the flag queue.   The answer does not seem to attempt to address the question at all. It looks like a comment on one of the other answers (probably Luca's).  It got rejected with the following reason:  declined - flags should not be used to indicate technical inaccuracies,            or an altogether wrong answer   I did not flag this because I thought it was wrong, I flagged it because I didn't think it was an attempt to answer the question. For what it's worth, I don't get what the guy who provided that answer even tried to say.  Was I wrong to flag it? If so why? What does the underscores in _servicename_ mean in AngularJS tests? Windows 10 on asus x44h touchpad scolling not working I have just updated Windows 7 to Windows 10.  My touchpad scolling (right-side) is not working anymore.    I have no idea how to fix this in Windows 10.  I have hit the start button and typed "synaptic", but have found nothing useful.  I used to have an icon on the taskbar that showed where I was touching the touchpad.  I used to be able to right-click that icon to change settings, but Windows 10 has not brought that icon over.  Also, the asus website has no synaptic touchpad drivers available for Windows 10.  I have tried the enaltech drivers, but upon the installation atempt I am told that my hardware is not correct. why would curl and wget result in a 403 forbidden? How to set the fully qualified hostname on CentOS 7.0? How do I set the fully qualified hostname on CentOS 7.0?  I have seen a few posts online for example using:  $ sudo hostnamectl set-hostname nodename.domainname   However, running domainname returns nothing:  $ domainname (none)   Also:  $ hostname nodename.domainname   However,  $ hostname -f hostname: Name or service not known $ hostname -d hostname: Name or service not known   Some debug output:  $ cat /etc/hostname nodename.domainname  $ grep ^hosts /etc/nsswitch.conf hosts:      files dns I have a brand new 32GB flash drive that was bought from Hong Kong but manufactured in China.  When I plug it in, it's detected, and when I examine the properties of the drive the size is reported to be 31.2 GB.  Windows forces me to format it but I have not manged to actually do that.  I then I browsed the internet and a forum told me to change the letter to L. After I did that I searched many programs that can help me format it, but when I checked it back, it is 4MB now.  Any idea what's happening with my drive? I want to search for what += means, but Google won't return any documents.  Also when I come across some PHP code I don't understand I can't search for the keywords as written.  How can I search for these special characters or include these special characters in a search. Hair particles render not same as 3dView(Eevee) Hair particle renders are not same as in 3dView(Eevee). Hair is using Principle Bsdf shader and  camera is set to Orthographic. Force and turbulence fields are applied to it. How to resolve this? VMWare: This host doesn't support VT. Can not play 64-bit VM due to lack of VT flag I have an Intel E7300 Core2 Duo processor with 2.66GHz speed. I need to run a 64-bit WinXP SP2 VMWare image. My host operating system is Win 7. So I just installed the VMPlayer from www.vmware.com.  The problem is when I run the VM image it says This host doesn't support VT.  If I continue I see Windows shows an error message saying     Attempting to load an x64 operating system, however this CPU is not compatible with x64 mode.   My question is how do I run this VM without buying a new processor? Any other tools or software? Before you disagree with me with a downvote, or just want to be funny, please hear me out. From the :  The upvote privilege comes first because that's what you should focus on: pushing great content to the top. Downvoting should be reserved for extreme cases. It's not meant as a substitute for communication and editing.  I have honestly never seen a downvote reserved for an extreme case. The process is usually 1) downvote post if it's bad and don't comment. (Not much of a process, is it?) As you stay on the site longer and longer, you develop a carelessness towards downvotes (as I have), but for new users, it's really discouraging.  question inspired me to [ask balpha to] make , which is a list of users with 1 rep who have the  badge. That's the closest I could get to new users who deleted their own posts after downvotes came in. Don't get me wrong, I know the argument about being nice to new users very well, but I still believe it to be an issue. And since , users are much more inclined to go the lazy route and downvote bad questions. No alternative measures (see below) are taken, for the most part. Personally, I leave a comment and don't downvote until the asker shows no interest in fixing their post. The only other measure taken to discourage downvotes (on answers) is the -1 rep punishment to the downvoter, which is absolutely meaningless and inconsequential anyways. How are we supposed to educate new users when we don't teach them what to do, don't direct them to the FAQ, and discourage them with downvotes right off the bat?  Instead of voting down: If the post is spammy or offensive — flag it. If the question is duplicate or off-topic — flag it for moderator attention. If something is wrong, please leave a comment or edit the post to correct it.  We claim to discourage downvoting, yet I see answers like  that encourage us to downvote first and ask questions later when a question is simply unclear or effortless. It's just an awful chain reaction. A new user comes to the site asking a question, but that doesn't work out too well, so they get downvotes, delete the question in shame, and don't get the help they deserve as a fellow writer of code and programmer. Strip away the quality of the site for a moment: the core functionality of this site, and the only reason I and a lot of other users are (still) on it, is to receive and give help to/from others. I enjoy making sure fellow programmers get the answer they need, because this site has helped me get mine when I needed them. Quality can still be maintained if content is fixed rather than downvoted more often. So can we stop discouraging new users from continuing to use this site and actually start to discourage downvoting? I think we need to encourage effort towards teaching and helping new users understand the system and why their post is bad rather than slapping them in the face with downvotes that they don't know how to interpret anyways. I'm leaving ways we can do this up to others. My first thought was to increase the punishment for downvoting answers and to no longer make question downvotes free, but those wouldn't be well-received, judging by the history of similar suggestions. . Why Doesn't C# Allow Static Methods to Implement an Interface? Why was C# designed this way?  As I understand it, an interface only describes behaviour, and serves the purpose of describing a contractual obligation for classes implementing the interface that certain behaviour is implemented.  If classes wish to implement that behavour in a shared method, why shouldn't they?  Here is an example of what I have in mind:  // These items will be displayed in a list on the screen. public interface IListItem {   string ScreenName();   ... }  public class Animal: IListItem {     // All animals will be called "Animal".     public static string ScreenName() {         return "Animal";     } .... }  public class Person: IListItem {      private string name;      // All persons will be called by their individual names.     public string ScreenName() {         return name;     }      ....   } Is it possible to set a global Blender preference so that all keyframed motion is linear by default? I've been working on some animations recently, and I've noticed that I'm having poor luck with certain things, namely the fact that I don't really use Blender's graph editor very well.  One of the major issues that I have is pretty widespread among 3D graphics programs; when I create an animated object, it defaults to using a curved graph instead of a linear graph, which means that the most movement is at the middle of the animation, and stuff just practically stops near the beginning or the end. While this helps with organic camera movements or such, it's also not what I usually want. Can I set animation curves to default to linear? I have 1-band GeoTIFF file. The color interpretation for band is gray. I want a color table in order to transform my Col.tiff to Col.jpeg or Col.png.  I tried putting a color table in this tiff via a .vrt:  gdal_translate -of VRT  Col.tiff Col.vrt   then adding in the .vrt:  &lt;ColorInterp&gt;Palette&lt;/ColorInterp&gt; &lt;ColorTable&gt;    &lt;Entry c1="0" c2="0" c3="255" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="254" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="253" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="252" c4="240"/&gt; &lt;/ColorTable&gt;   And translate again :  gdal_translate Col.vrt Col.tiff   But it answered:  Warning 1: Unable to export color table to GeoTIFF file.  Color tables can only be written to 1 band or 2 bands Byte or UInt16 GeoTIFF files.   How can I do it?  For information, there is gdalinfo for the Band:  Band 1 Block=55x18 Type=Float64, ColorInterp=Gray   Description = Something   Metadata:     GRIB_COMMENT=Something[p]     GRIB_ELEMENT=TMP     GRIB_FORECAST_SECONDS=0 sec     GRIB_REF_TIME=1401926400 sec UTC     GRIB_SHORT_NAME=SMTHG     GRIB_UNIT=[p]     GRIB_VALID_TIME=1401926400 sec UTC Maximum length of a USB Cable? I have a set of polygons representing administrative districts.  The districts were digitized by hand, and there are small spaces (slivers) between polygons where the polygons should be touching. How can I have the polygons snap together, removing the spaces? How to check if a 2D point is in a polygon?