$\sum _{k = 0}^ n \binom ka \binom {n-k}b = \binom {n+1}{a+b+1} $ Can anyone please give me a hint to prove it using combinatorics ? I know that $n \geq a+b$ and $k$ will run from $a$ to $n-b$ . But I think it should be $\binom {n}{a+b} $. Can anyone please help me?
Given two positive integers $a$ and $b$, prove that  $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$  I think there should be a good combinatorial proof for this, given the simplicity of the right-hand side... my hunch is to form all possible subsets of size $a+b+1$ by dividing the set into $k$ and $n-k$ elements, then choosing $a$ from the first, $b$ from the second, iterating over all values of $k$.  I can't quite make this work, so I've also been trying to rewrite ${n+1 \choose a+b+1}$ to make it clear.  Open to any proof method, all help appreciated.
Are security.ubuntu.com updates eventually merged into normal updates? Upon my observation, seems new packages will first appear in security.ubuntu.com, but if you upgrade later, they will appear in XX.archive.ubuntu.com (depends on your config),  Was that true ? Can not find the tag library descriptor for “http://java.sun.com/jsp/jstl/core” Is any aspect of the supernatural testable? What level of proof is possible for the supernatural? Simple & easy way to jail users I need a simple and easy way to jail users in their home directories in Oneiric. Do you have a simple configuration for jailing users, with full help, or some good web links?  I would be offering an online free public server with 10 to 20 GB free space. I don't know how many users. I want to give them SSH and SFTP so that they can connect through FileZilla. Very simply put:  I have a class that consists mostly of static public members, so I can group similar functions together that still have to be called from other classes/functions.  Anyway, I have defined two static unsigned char variables in my class public scope, when I try to modify these values in the same class' constructor, I am getting an "unresolved external symbol" error at compilation.  class test  { public:     static unsigned char X;     static unsigned char Y;      ...      test(); };  test::test()  {     X = 1;     Y = 2; }   I'm new to C++ so go easy on me. Why can't I do this? Show that every connected graph with $n$ vertices has at least $n − 1$ edges.  How can I prove this?  Conceptually, I understand that the following graph has 3 vertices, and two edges:  a-----b-----c  with $a$, $b$ and $c$ being vertices, and $\{a,b\}$, $\{b,c\}$ being edges.  Is there some way to prove this logically?  --UPDATE--  Does this look correct?  Any advice on how to improve this proof would be appreciated.  Thank you. Proof of Heron's Formula for the area of a triangle Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula: $$A = \sqrt{p(p-a)(p-b)(p-c)},$$ where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$.  Could you please provide the proof of this formula? Thank you in advance. I have a default Blender cylinder positioned upright atop a ground plane:    How can I rig the cylinder so that can be animated to wiggle/jitter (perhaps procedurally with noise f-curve modifiers on two of its rotation axis) without penetrating the floor?    A simple parent-child relationship as in the image above is sufficient to rotate (and translate) the cylinder on 1 axis in either a positive OR negative direction.   I'm looking for a setup that would allow me easily to rotate the cylinder on two or even three axis in both a positive and negative direction.  I imagine this would be possible with some drivers applied to the XYZ coordinates using trigonometric equations but the math escapes me. It might also be possible using a particular bone setup with the correct constraints but my attempts to find something that works so far have not succeeded.  Thanks for your assistance. Otherwise, is there any alternative command line utility that can achieve this? Spaces in files names in \input or \includegraphics I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this? Can I insert an image into an equation? I would like to use a small image in an equation, where a symbol like \alpha might otherwise be. Is there a nice way to do this? Concrete Mathematics - The Josephus Problem I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$? URLs: Should I use hyphens, underscores or plus symbols? Which is better for search engines?   example.com/my_cool_page.html example.com/my-cool-page.html  example.com/my+cool+page.html How do I write a good title? A good title helps your question get the attention it deserves.  What goes into a good title? Finding net magnetic and electric force on charged particle This is from my textbook, it is not an assigned problem, but I want to understand.   It says:      Consider the situation in the figure, in which there is a uniform electric field in the x direction and a uniform magnetic field in the y direction. For each example of a proton at rest or moving in the x, y, or z direction, what is the direction of the net electric and magnetic force on the proton at this instant?   I believe I need to use the equation   $$ F_{net}=(q\overrightarrow{E})+q(\overrightarrow{v} \times \overrightarrow{B}) $$  But I'm not sure exactly how. We've just started to learn about this, and I want to get a head start. Could anyone put me on the right track? Peer to Peer: Methods of Finding Peers Are there any known methods of finding peers without using a dedicated central server?  ie: If I have peers which are disconnecting and reconnecting to the internet but getting a new IP address each time, and I want to connect to them without setting up a dedicated server to register with.  I was thinking about using peers email address to send a manifest of connected peers periodically, with some sort of timecode, negating the need for a dedicated server.  This would be a fallback if none of the peers could be connected to after trying all the previously known peer addresses.  But existing models of finding peers would be preferable. Ability to give away iOS application, usually for sale I am wondering if it's possible to do "mass giveaways" on the AppStore, without lowering, changing or otherwise changing the listed application.  Is it possible to distribute hundreds to, potentially, thousands of promo codes, legally, while still leaving the "normal" distribution channel of purchasing the app open?  An example of the above would be selling physical items in a store which have a bundled digital component available. Essentially you buy a product in a store, which comes "bundles with a free app" (BB and Android have already been dealt with, iOS is unclear). It would be nice to give the option to have a promo code in the box, that the user can then use to get the application for free. Is this possible? Center of a free product How to secure my laptop so that hacking by physical access is not possible?