Combinatorial proof $\sum_{i=k}^n {i-1 \choose k-1} = {n \choose k}$
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
Ok guys, continuing my passage through edwards... here is the question... thanks for hints/solutions in advance:  Suppose $f(x,y,z)=0$ can be solved for each of the three variables $x,y,z$ as a differentiable function of the other two. Then prove that  $\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$  Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant). What is the definition of a timelike and spacelike singularity? How does a half-life work? Carbon-14 has a half-life of 5,730 years.  That means that after 5,730 years, half of that sample decays.  After another 5,730 years, a quarter of the original sample decays (and the cycle goes on and on, and one could use virtually any radioactive isotope).  Why is this so?  Logically, shouldn't it take 2,865 years for the quarter to decay, rather than 5,730? Is there a location where I can drop in a custom JavaScript file in Drupal 7.x?  It just includes simple event actions for menus, and I don't want to edit the theme/module files.  Or do I have to just add the "include" hook to the theme/module? Does anyone know the name of travel posters from the 1930s? Is there a name for this design style - use of flat colours for dimension and shading - that was so prevalent in UK travel posters of the 1930s? What else can you do in the turn you ready an action? Is a block of cotton heavier than a block of iron? if i have a block of cotton and a block of iron each containing a weight of 100 kg when measured on a weighing machine, in reality which one is heaver than the other, which one is heavier and why?  This question was in my physics worksheet. The teacher gave the following answer: the apparent weight of both of them is the same, but the buoyant force of cotton is high since the volume of cotton is more than iron. Therefore, true weight of cotton is more. Does this answer make sense? What is the right answer? Assume $0 &lt; a_0 \leq a_1 \leq \cdots \leq a_n$, show that equation   $$ p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0 $$   does not have root in $|z| &lt; 1$. Here $z$ is complex.   I started by considering the case in which $n = 2$, and tried to use the triangle inequality, while no progress was made. Any hints will be appreciated. Run true multiple process instances of gnome-terminal (Posted here as probably applies to all sites).  I was examining my reputation gain yesterday on Stack Overflow: 190. I believe I should have hit the 200 cap. Mainly due to         Respectively, I had 10, 7, 3, and 2 upvotes on those questions alone. There are a couple of other minnows in the list but the above alone should enable me to hit the cap.  There was a user removed sometime yesterday which knocked off 10 points (and possibly accounts for the disparity) but I think (1) the cap should have been recalibrated and (2) I answered (3) after this event before turning in for the night.  I checked all the usual suspects: the reputation graph on your profile and  the achievements tab, but they are all showing 190. Yes, it's time for me to complain about plagiarism again.  We already have Suggested Edit audits, but I would like to see Tag Wiki and Tag Wiki Excerpt audits that specifically introduce copied content for that tag without including the appropriate source(s), and warn people who approve it that they just approved a tag with plagiarized text.  For example, an audit for the  tag could be:     Microsoft SQL Server is a relational database management system developed by Microsoft. As a database, it is a software product whose primary function is to store and retrieve data as requested by other software applications, be it those on the same computer or those running on another computer across a network (including the Internet). There are at least a dozen different editions of Microsoft SQL Server aimed at different audiences and for different workloads (ranging from small applications that store and retrieve data on the same computer, to millions of users and computers that access huge amounts of data from the Internet at the same time). Its primary query languages are T-SQL and ANSI SQL.   Anyone who spends 15 seconds Googling this will see where it came from -- the problem is, most people don't.  I know with Suggested Edit Audits, it's just random text to catch the most egregious reviewers; for this, you would have to most likely select a small sample of tags, and then create their copy-and-paste entries for audits.    The whole point of this would be to raise awareness for people who have figured out how to recognize and pass Suggested Edit audits, but still don't understand the finer nuances of reviewing. Instant e-mail notifications of answers to questions I'd like to have Instant notifications by email of answers to my questions.  Can this be implemented? What is the max variance distribution with fixed mean and support Is there a upper bound on the variance for a distribution that is confined say, in $[0,1]$, and has fixed mean $\mu \in [0, 1]$ How to add a realistic flare How do I turn the little emission sphere into a realistic flaring sun. (This piece is animated with all the planets rotating and revolving) How to make left-right transition of overlay image ffmpeg What component to use for audio signal switching? I'm looking to build a guitar pedal switcher based on a Raspberry Pi or Arduino (have not yet decided, but I have an unused Raspberry around and I lean towards that). Anyway, digital logic.  Now, I need to switch audio signal obviously and I need to do it quietly (both physical silence and electrical silence). Please note that the signal is line level throughout most of the build, except for the input which is much lower. I will probably have 2 or 3 loops before the preamp so those may not encounter line level signal.  From what I read and know, so far, mechanical relays are not really an option due to the clicking noise they make. I also read something about FETs, but that's a bit out of my league and I would appreciate any info on these.  PS: I will probably use different power rails for the digital and analog paths.  PS2: The whole thing will be mounted in a 1U rack enclosure, so please tell me if I should take into consideration any thermal issues. Can I manually limit the %CPU used by a process? Difference between Login Shell and Non-Login Shell? I understand the basic difference between an interactive shell and a non-interactive shell.  But what exactly differentiates a login shell from a non-login shell?  Can you give examples for uses of a non-login interactive shell? Are there any best practices for choosing the title of web pages? There are a lot of resources online oriented toward SEO, but not toward users.  My impression is that most sites have something similar to Product - Page. But there are cases when switching this around is more useful. For example, if mail clients displayed the title like this:  Inbox (3) - Gmail  Instead of:  Gmail - Inbox (3)  You would be able to see the message count, even if the title was cut off. (I'm pretty sure Gmail used to do this).  I assume the title is mostly ignored by users except when:   using boorkmarks finding the right tab in the browser   What are the considerations for users when choosing a web page title? Are there different considerations for the home page, pages behind a login, etc? So I had  closed within eight minutes of asking.  Is this a little fast? Not only do I object to the reason it was closed (not a real question, come on I'm asking for decent books!), but it was closed insanely fast. Should there be a grace period for questions (diamond mods exempt if so) before they can be voted on to be closed?