$$\binom{r}{r} + \binom{r+1}{r} + \binom{r+2}{r} + \dotsb + \binom{n}{r} = \binom{n+1}{r+1}$$  I don't have much experience with combinatorial proofs, so I'm grateful for all the hints.  (Presumptive) Source: Theoretical Exercise 1.11, P18, A First Course in Pr, 8th Ed, by S Ross
Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but:  $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$  for some integers $m, n$  $$\implies \left(\sqrt[3]{a} + \sqrt[3]{b}\right)^3 = \frac{m^3}{n^3}$$  $$\implies a + b + 3 \cdot \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right) = \frac{m^3}{n^3}$$  Since $a +b$, $\sqrt[3]{a} + \sqrt[3]{b}$ are rational, $\sqrt[3]{ab}$ must be rational as well.   For convenience let us say $\sqrt[3]{a} = p, \sqrt[3]{b} = q \implies pq$ is rational. This means, for all $i$, $p^iq^i$ is rational.  $$\implies (p + q)^2 = \frac{m^2}{n^2}$$  $$ = p^2 + q^2 + 2pq = \frac{m^2}{n^2}$$  Since $pq$ is rational, $2pq$ is rational and so is $p^2 + q^2$.  Assume, for some $i$ that $p^i + q^i$ and $p^{i-1} + q^{i-1}$ is rational.   $$\implies (p^i + q^i)(p + q) - pq(p^{i - 1} + q^{i - 1}) = p^{i+1} + q^{i+1}$$  is rational as well.  So for all $i$,  $$a^{\frac{i}{3}} + b^{\frac{i}{3}}$$  and   $$a^{\frac{i}{3}}b^{\frac{i}{3}}$$  are rational.  I know I'm really close to the answer, but it somehow just keeps slipping through my fingers. Problem installing .deb in Software Install Ubuntu 20.04 I tried to install some .deb files in Ubuntu 20.04 from GitHub and other internet sources, but when I open it with Software Install it gives me  problem (Failed to install file: not supported). The only way for me to install them is to go to the downloaded file and do sudo dpkg -i file.deb. Can anyone help? I don't know what to do and I don't want to use the terminal every time. Any help would be appreciated, thanks EDIT: that's NOT a problem with .deb in Ubuntu, it is a problem with Software install. I can install every .deb through the terminal but not with Software install, and Ubuntu suggests to install .deb with Software install. That's not a dependencies problem or anything similar, that's an issue with installing .deb WITH Software install and I would like to know how I can fix Software installer Lorentz Invariant Integration Measure Conformally Map Region between tangent circles to Disk do { ... } while (0) — what is it good for? Possible Duplicate:          I've been seeing that expression for over 10 years now. I've been trying to think what it's good for. Since I see it mostly in #defines, I assume it's good for inner scope variable declaration and for using breaks (instead of gotos.)  Is it good for anything else? Do you use it? Mount USB drive with write permissions for everyone or specific user How to check the maximum number of allowed connections to an Oracle database? Proving that the tangent to a convex function is always below the function Configure Postfix to use external MX servers for delivery of local mail if user is unknown Does the OP of a flagged post get comment notifications if the post is deleted by mods? How to make beveled 90 degree corners on a curve better? I know the title is a mouthful, but this is the problem: I made a curve with 90 degree corners, like so:  I then beveled it, and this was the result:  As you can imagine, this isn't very pretty. I'm looking for something like this:  How can I achieve this look? How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. Why does python 2.5.2 have the following behavior  &gt;&gt;&gt;[2].extend([]) == [2] False  &gt;&gt;&gt; [2].extend([]) == None True  $ python --version Python 2.5.2   I assume I'm not understanding something here, but intuitively I'd think that [2].extend([]) should yield [2] Installing an addon There is this addon that was built called Window Generator 2. It can be . However, when I go to the download link it shows up in a new tab as a bunch of code.  How can I download and use this addon in Blender? Introductory Calculus textbook for physics? I'm Egyptian and flying to Cairo from Tbilisi, via Istanbul. The first flight Tbilisi-Istanbul is on Atlasjet, then from Istanbul to Cairo I am flying on Egyptair. For that, I have to recheck in at Istanbul to my Egyptair flight  Will I need to pass passport control at Istanbul airport to collect my bags and check in on Egyptair? Will I need a transit visa for that? Or it can be all done through neutral zone? Is there a proof from the first principle that the Lagrangian L = T - V? Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. I am using nginx/0.7.68, running on CentOS, with the following configuration:  server {     listen       80;     server_name ***;     index index.html index.htm index.php default.html default.htm default.php;      location / {             root   /***;             proxy_pass   http://***:8888;             index  index.html index.htm;     }     # where *** is my variables   The proxy_pass is to a DNS record whose IP changes frequently. Nginx caches the outdated IP address, resulting in a request to the wrong IP address.  How can I stop nginx from caching the IP address, when it is outdated? Catch multiple exceptions at once? It is discouraged to simply catch System.Exception. Instead, only the &quot;known&quot; exceptions should be caught. Now, this sometimes leads to unnecessary repetitive code, for example: try {     WebId = new Guid(queryString[&quot;web&quot;]); } catch (FormatException) {     WebId = Guid.Empty; } catch (OverflowException) {     WebId = Guid.Empty; }  I wonder: Is there a way to catch both exceptions and only call the WebId = Guid.Empty call once? The given example is rather simple, as it's only a . But imagine code where you modify an object multiple times, and if one of the manipulations fails expectedly, you want to &quot;reset&quot; the object. However, if there is an unexpected exception, I still want to throw that higher. Pop up option are not popping up after snipping the area