Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
How to find/grep what is between string1 and string2? I am trying to extract a value from a long string that may change over time. So for example the string could look something like this  ....../filename-1.9.0.3.tar.gz"&lt;....   And what I want to extract is the value between filename- and .tar.gz, essentially the file version (1.9.0.3 in this case). The reason I need to do it this way is because I may later run the command and the value will be 1.9.0.6 or 2.0.0.2 or something entirely different.  How can I do this? I'm currently only using grep, but I wouldn't mind using other utilities such as sed or awk or cut or whatever. To be perfectly clear, I need to extract only the file version part of the string, since it is very long (on both sides) everything else needs to be cut out somehow. babel shorthand "| doesn't work in macros The package babel with the ngerman option (and most other languages as well, but not english) provides the shorthand "|, which prevents a ligature at the given position (see p. 81 of the , e.g. Auf"|lage will result in the word Auflage without an fl-ligature (as is desired because, in German, you don't want ligatures across morpheme borders).  However, when placed in a macro, "| doesn't work anymore. Why is this and how can I circumvent it?  \documentclass{article}  %\usepackage[T1]{fontenc}% doesn't help \usepackage[ngerman]{babel}  \newcommand{\foo}{"|}  \begin{document}  ff  f"|f  f\foo f  \end{document} Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $. Publish and/or subscribe to Platform Events between separate Orgs? My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse.  But my approach requires the associativity of function composition.  Let $f:  X \rightarrow Y, g:  Y \rightarrow Z, h:  Z \rightarrow W$ be functions.  $((f \circ g) \circ h)(x) = h((f \circ g)(x)) = h(g(f(x)))$, and $(f \circ (g \circ h))(x) = (g \circ h)(f(x)) = h(g(f(x)))$.  However, I am having problems in justifying that the two compositions, $(f \circ g) \circ h$ and $f \circ (g \circ h)$, have the same domain and range.  When I consulted ProofWiki, whose link is at the bottom, I got even more confused.  Specifically, for $(f \circ g) \circ h = f \circ (g \circ h)$ to be defined, ProofWiki requires that dom$g =$ codom$f$ and dom$h =$ codom$g$.  First of all, I think that it should be dom$g =$ range$f$ ....  Moreover, as you can see in the example below, you actually have to adjust domains and ranges of $f, g, h$ for the requirement to hold true.  Let $f: \mathbb R \rightarrow \mathbb R$ be $f(x) = 2x$, $g: \mathbb R^+ \rightarrow \mathbb R$ be $g(y) = ln(y)$, $h: \mathbb R \rightarrow \mathbb R$ be $h(z) = z - 10$.   Then $((f \circ g) \circ h)(x) = ln(2x) - 10 = (f \circ (g \circ h))(x)$, with dom$((f \circ g) \circ h) = \mathbb R^+$ = dom$(f \circ (g \circ h))$.  As a result, we need to set dom$f = \mathbb R^+$, range$f = \mathbb R^+$; dom$g$, range$g$, dom$h$, and range$h$ remain the same.  Am I allowed to do that?  This adjustment implies that when we say dom$f = X$, $f$ must be defined for all elements in $X$, but $X$ may not be the entire set of elements for which $f$ is defined. Sometimes I wonder if he is actually/really  as lavish as he   pretends to be.   I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be? Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the identity; hence, $r &gt; 1$. If $r = 2$, then we are done. Suppose that $r &gt; 2$. In this case the product of the last two transpositions, $τ_{r−1}τ_r$, must be one of the following cases: $(ab)(ab) = id\\ (bc)(ab) = (ac)(bc)\\ (cd)(ab) = (ab)(cd)\\ (ac)(ab) = (ab)(bc),$ where $a$, $b$, $c$, and $d$ are distinct. The first equation simply says that a transposition is its own inverse. If this case occurs, delete $τ_{r−1} τ_r$ from the product to obtain $id=τ_1τ_2···τ_{r−3}τ_{r−2}$. By induction $r − 2$ is even; hence, $r$ must be even. In each of the other three cases, we can replace $τ_{r−1} τ_r$ with the right-hand side of the corresponding equation to obtain a new product of $r$ transpositions for the identity. In this new product the last occurrence of $a$ will be in the next-to-the-last transposition. We can continue this process with $τ_{r−2} τ_{r−1}$ to obtain either a product of $r − 2$ transpositions or a new product of $r$ transpositions where the last occurrence of $a$ is in $τ_{r−2}$. If the identity is the product of $r − 2$ transpositions, then again we are done, by our induction hypothesis; otherwise, we will repeat the procedure with $τ_{r−3}τ_{r−2}$. At some point either we will have two adjacent, identical transpositions canceling each other out or $a$ will be shuffled so that it will appear only in the first transposition. However, the latter case cannot occur, because the identity would not fix $a$ in this instance. Therefore, the identity permutation must be the product of $r − 2$ transpositions and, again by our induction hypothesis, we are done. Question: That is the proof written in the book, but unfortunately after more than $10$ times reading I don't understand the last two paragraph (i.e., from &quot;In each of the other three cases&quot;). Is there another simpler proof or could someone please clarify what is going on? Thank you. "Abusing" bounties for unlimited profit Six days ago I put a +50 bounty on  which wasn't asked by me, but to which I'd previously posted a pretty decent answer. Since then I've earned 130 reputation from upvotes on my answer, netting me +80 reputation and a bronze badge from no work.  It seems possible to slowly generate free reputation by "investing" in your answers like this. I expect I could repeat this perpetually and continue to profit from it.  This strikes me as an abuse of the bounty system. Should anything be done to prevent it?  EDIT: For more concrete questions:   If I put another bounty on this question after this one expired, would that be considered abuse? What if I did it several times? ( the answer was  "yes", but the author was doing so to promote his own project.) What if I didn't do it to the same question, but was constantly promoting questions I'd answered, in order to promote my answers? I am travelling from Singapore to San Francisco and the airline has a layover of 7.5 hours at Tokyo Airport (NRT). I am wondering if I would be allowed to go out to experience the city for a while in some form a visitor visa.  I know Singapore has such facilities, but does anyone know if I can do it in Tokyo? Any suggestions or pointers are welcome. Note - I do not have a visa for Japan yet, but I am hoping that 1 day visitor visa could be possible at airport. How do you tell what brand/make/model of bike I have? PHP import Excel into database (xls & xlsx) Magnitude of standardized coefficients (beta) in multiple linear regression Being aware of , I am curious about the question how big standardized coefficients can get. I had a discussion with my professor about that issue and she was arguing standardized coefficients (beta) in multiple linear regressions can not become greater than |1|. I have also heard that predictors with standardized coefficients greater than 1 should not be be included/appear in multiple linear regression. When I recently estimated a multiple linear regression in R using lm(), I estimated the standardized coefficients with lm.beta() function from the package 'lm.beta'. In the results I could observe a standardized coefficient greater than one. Right now I am just not sure about what is the truth.  Can standardized coefficients become greater than |1|? If yes, what does that mean and should they be excluded from the model? If yes, why?  I would be very thankful, if somebody could make this issue clear for me. SQL Server 2000 on Windows 10 I need to install SQL Server MSDE 2000 SP4 on Windows 10.  I have other machines that run Windows 7 and 8.1 and SQL Server MSDE 2000 works fine (with merge replication configured too).  Now I testing the same scenario with Windows 10, but setup is closing unexpectedly during installation.  I know that this version is very old, but is there a workaround to install it on Windows 10? Deleting a middle node from a single linked list when pointer to the previous node is not available Is it possible to delete a middle node in the single linked list when the only information available we have is the pointer to the node to be deleted and not the pointer to the previous node?After deletion the previous node should point to the node next to deleted node. I want to remove all empty lines from a file. Even if the line contains spaces or tabs it should also be removed. Enable SSO for Salesforce Marketing Cloud I need to enable Single Sign On for Salesforce Marketing Cloud account where Identity Provider will be a third party and Marketing Cloud will be the Service Provider. I already got SSO Metadata option enabled for my account. I however, need an Entity ID and Federation ID. I am not clear about how to carry the entire process. If anyone of you guys has done SSO enabling for Marketing Cloud, please share steps. Where do the electrons come from when an electric generator is making electricity?  Is from the air?  Would a generator work in a vacuum?  Electrons have mass so where would they be pulled from if there is nothing? When is tension constant in a rope? Consider a massless rope with pulling forces applied at each end.  How do we decide if tension is constant or not in a rope? Consider a few example scenarios:  For example, if there is a knot in the rope the tension is not constant throughout (why?. Similarly, if the rope is hung over a cylindrical pulley of a non-negligible radius, the tension is not constant (why?). If there isn't anything touching the rope, for example, 2 people tugging on the rope at each end, the tension is constant (why?). Are the proofs for the properties of parallel lines, and that a triangle has 180 degrees, inherently tautological? I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance. Getting disk i/o statistics for single process in Linux