Find the coefficient of ${x^9}$ in the expansion of $\left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^3}} \right)..\left( {1 + {x^{100}}} \right)$. The official answer is 8. How do I find the general term, Dividing the above equation by $(1-x)$ is not generating the required result.
The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)...(1+x^{100})$.  I tried the following concept, how to sum 9 using 1-9 only without repetition.  1:9, 2:1+8,3:7+2, 4: 6+3, 5: 5+4,  6:1+2+6 ,7:1+3+5, 8:2+3+4  The answer is 8.  How will it be solved using Binomial theorem.
What was the origin of Star Trek's Borg? SHA-256 "almost unique"? I have seen numerous references on the internet of people describing SHA-256 as generating an "almost unique" hash.  . there are more.  Is there some mathematical basis to the almost uniqueness? I would have thought that as SHA-256 is an evenly-distibuted (?) hash algorithm which can accept any message (infinitely many) as an input and produces a fixed length (finite) hash - then there will in fact be an infinite number of each possible output - which I would think hardly qualifies as "almost unique".  Is there a basis for this statement? What's the most popular way to encrypt individual files or folders? Is there a way to refresh Notepad? Say, for example, that there's one process that's writing to a file. While the process is running, I opened the file in notepad. The process keeps  writing to the file.   Other than closing and reopening the file, is there any way for me to "refresh" the data that notepad is showing? Get a resource using getResource() I was wondering what the official term for the relationship described below is.   So My Grandparents D &amp; F had two sons: T &amp; E. T is my father. E married L and had three kids: P, E, &amp; A which are my cousins.    Now L has a sister S (P,E, &amp; A's Aunt) who has two kids C &amp; B (P, E. &amp; A's Cousin).   Now what is the official term for C &amp; B in relation to me? How to create a SQL Server function to "join" multiple rows from a subquery into a single delimited field? To illustrate, assume that I have two tables as follows:  VehicleID Name 1         Chuck 2         Larry  LocationID VehicleID City 1          1         New York 2          1         Seattle 3          1         Vancouver 4          2         Los Angeles 5          2         Houston   I want to write a query to return the following results:  VehicleID Name    Locations 1         Chuck   New York, Seattle, Vancouver 2         Larry   Los Angeles, Houston   I know that this can be done using server side cursors, ie:  DECLARE @VehicleID int DECLARE @VehicleName varchar(100) DECLARE @LocationCity varchar(100) DECLARE @Locations varchar(4000) DECLARE @Results TABLE (   VehicleID int   Name varchar(100)   Locations varchar(4000) )  DECLARE VehiclesCursor CURSOR FOR SELECT   [VehicleID] , [Name] FROM [Vehicles]  OPEN VehiclesCursor  FETCH NEXT FROM VehiclesCursor INTO   @VehicleID , @VehicleName WHILE @@FETCH_STATUS = 0 BEGIN    SET @Locations = ''    DECLARE LocationsCursor CURSOR FOR   SELECT     [City]   FROM [Locations]   WHERE [VehicleID] = @VehicleID    OPEN LocationsCursor    FETCH NEXT FROM LocationsCursor INTO     @LocationCity   WHILE @@FETCH_STATUS = 0   BEGIN     SET @Locations = @Locations + @LocationCity      FETCH NEXT FROM LocationsCursor INTO       @LocationCity   END   CLOSE LocationsCursor   DEALLOCATE LocationsCursor    INSERT INTO @Results (VehicleID, Name, Locations) SELECT @VehicleID, @Name, @Locations  END      CLOSE VehiclesCursor DEALLOCATE VehiclesCursor  SELECT * FROM @Results   However, as you can see, this requires a great deal of code.  What I would like is a generic function that would allow me to do something like this:  SELECT VehicleID      , Name      , JOIN(SELECT City FROM Locations WHERE VehicleID = Vehicles.VehicleID, ', ') AS Locations FROM Vehicles   Is this possible?  Or something similar? Is 225.225.225.128 a valid subnet mask? DIY.Blogoverflow.com is redirecting to stackexchange.com/blogs I seem to be the unlucky recipient of a 302 redirect from the blogoverflow site, and was asked by Grace Note to upload the issue here. While trying to access the DIY blog site, I'm getting bounced over to the generic SE blog page. I've tried resolving this by changing DNS servers, clearing my browser cache, and trying a different browser. Others going to the same site aren't experiencing the issue, but I'm at a loss for what's different on my system that would cause this. Here's the wget output of the redirect:  --2014-02-18 14:59:28--  http://diy.blogoverflow.com/ Resolving diy.blogoverflow.com (diy.blogoverflow.com)... 67.228.172.87 Connecting to diy.blogoverflow.com (diy.blogoverflow.com)|67.228.172.87|:80... c onnected. HTTP request sent, awaiting response...    HTTP/1.1 302 Found   Date: Tue, 18 Feb 2014 19:59:28 GMT   Server: Apache   X-Powered-By: PHP/5.4.22   Location: http://stackexchange.com/blogs   Content-Length: 1001   Connection: close   Content-Type: text/html; charset=UTF-8 Location: http://stackexchange.com/blogs [following] --2014-02-18 14:59:28--  http://stackexchange.com/blogs Resolving stackexchange.com (stackexchange.com)... 198.252.206.140 Connecting to stackexchange.com (stackexchange.com)|198.252.206.140|:80... conne cted. HTTP request sent, awaiting response...    HTTP/1.1 200 OK   Cache-Control: private   Content-Type: text/html; charset=utf-8   X-Frame-Options: SAMEORIGIN   Date: Tue, 18 Feb 2014 19:59:28 GMT   Content-Length: 16775 Length: 16775 (16K) [text/html] Saving to: ‘index.html.1’       0K .......... ......                                     100% 1.04M=0.02s itoa() is a really handy function to convert a number to a string. Linux does not seem to have itoa(), is there an equivalent function or do I have to use sprintf(str, "%d", num)? After the training phase, is it better to run neural networks on a GPU or CPU? My understanding is that GPUs are more efficient for running neural nets, but someone recently suggested to me that GPUs are only needed for the training phase. Once trained, it's actually more efficient to run them on CPUs.  Is this true? Why can't I use Stack Exchange if I'm under 13 years old, or if I'm under 16 years old and from the EU? How to hide "internal" polygon boundaries? Are there any guidelines for converting cooking recipes from Regular Ovens to Convection? I was looking to buy a Convection Oven and wanted to know if there are any general differences in baking methods between the two. $n$ Lines in the Plane How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines have a common point. Further, this is not a non-Euclidean problem, but I wouldn't mind a discussion on the non-Euclidean nature of the problem.  I was thinking it might be easier to show using the unit sphere.    A GOOD REFERENCE I FOUND THAT WILL BE USEFUL TO ANYONE COMING ACROSS THIS PROBLEM IS Concrete Mathematics, Graham, Knuth, Patashnik pp. 4–8. How can I use Gmail (in Chrome) as the default email handler in Windows 8?  I used to use GmailNotifier in Windows 7, but that doesn't seem to work anymore. Clicking on a mailto: link simply opens Chrome with a blank page.  I am not interested in using the Mail app built into Windows 8. When I updated from 14.04 to 14.10 I noticed my track-pad cursor and left and right mouse clickers were not working. I have tried looking this issue up and can not find any solutions anywhere. If anyone could help me that would be amazing, I am using a wired mouse to my USB port for a temporary fix. I have a Dell Inspiron 15R. How to move a database from SQL Server 2012 to SQL Server 2005 What are my options if I need to move a database from SQL Server 2012(32bit) to SQL Server 2005(64bit)?  I know I cannot:   restore a backup of the database on SQL Server 2005 detach &amp; attach   I know I can:   use import data wizard, and I tried it on one database, but it only moves data, and even that was troublesome as I needed to do a lot of work creating temp tables to maintain identity columns, recreating all FKs, indexes etc.    Is there an easier option? Default poster size What is the typical size for a conference poster? Is it "ask the organizers", or can I safely assume that I'll get a default size, and that there will be enough space to hang it at the poster session? The PHB is a bit unclear when it comes to readied spells in combat. If I lose my concentration do I lose the spell slot? What about if the trigger never occurs, do I get the spell slot back?