Combinatorics summation proof involving binomial coefficients
Show this equality (The factorial as an alternate sum with binomial coefficients).
Is there an infinite amount of wavelengths of light? Is the EM spectrum continuous? The electromagnetic spectrum is a continuum of wavelengths of light, and we have labels for some ranges of these and numerical measurements for many.  Question: Is the EM spectrum continuous such that between two given wavelengths (e.g. 200nm and 201nm) there is an infinite number of distincts wavelengths of light?  Or is there some cut-off of precision with which light might exist (e.g. can light only have wavelengths of whole number when measured in nanometers, etc.)? My &quot;hat&quot; is shown twice in my answer, once in my sig, and again above. MacOS 10.14.6; Safari Version 14.0.2 (14610.3.7.1.10) How to install Eclipse? I'm running macOS 10.15 Catalina Beta.  When I try to install Spotify I see the following error:    Is this a Spotify issue or a Catalina issue?  Thanks. 80s/90s book where boy stumbles on game where aliens gamble their planets I don't remember much about it other than the setting. It's about a boy who one night stumbles upon some different alien races playing a game in which their civilizations will end if they lose. He gets involved in the game somehow and is hunted. How can I find the XY coordinates of an HTML element (DIV) from JavaScript if they were not explicitly set? Lightning Web Component embedded in Visualforce Page via Aura Component and App not displaying I am writing a lightning web component that is going to be used in Salesforce Classic, Lightning Experience and an Employee Community. It does not HAVE to be a lightning web component - I could use a traditional Aura Lightning Component if I must.   However, what I am finding is that the visualforce page does not work, even when the lwc component is embedded in a aura component that is then used with Lightning Components for Visualforce. I can see from the documentation  that using a lwc directly from within visualforce is not supported, but I was under the impression that it may work if wrapped in an aura component.   My question then is this:  Has anyone had success having a lwc wrapped in an aura component display successfully within Visualforce? Or is this completely unsupported behaviour? Describe the units in $\mathbb{Z}[\sqrt{-d}]$ When Frodo put on the One Ring, both the Nazgul and Sauron himself found Frodo. However when Bilbo had the ring and possibly used it many times, Sauron sought Gollum to get a hint on where to look for the ring and send the Nazgul after it.   Sauron sought Gollum before Bilbo wore the ring at his birthday party, slightly after that time, he was very effective at locating Frodo when Frodo put on the ring. So I guess it's not a matter of Sauron gaining any extra powers with time.  Also Bilbo seemed to use the ring without it bothering him at any point, unlike Frodo who seemed to be very troubled when using the ring. How is that possible? How are self-replicating mines supposed to work? Why won't SparseArray let me store values with the head List? In-universe explanation for single line of planetary entry/departure in Star Wars Why are planetary blockades confined to a single point of a planet instead of being sphere shell shaped around the planet? (I know, it would take an awful lot of spaceships.)  Also, when a rebel ship wants to escape from a planet and there is a single star destroyer hovering in the atmosphere (or some more of them further up, like in the Empire Strikes Back) how come it's not a viable maneuver to fly low altitude for a couple of hundred kilometers horizontally (they would be out of the single-point blockade), then make a vertical dash upwards? Notebook computer battery says: plugged in but not charging with no further information. Is the battery bad already? My 1.5 year old Dell Studio 15 notebook computer has a battery and I keep it attached to the notebook but 98% of the time I just plug in the power adapter.  Lately when I unplugged the power adapter, the battery was only good for 10 minutes, and a few days ago, when I unplugged the power adapter, the computer simply turned off.  When clicking on the battery icon on Win 7, it says "Plugged in, not charging":    is the battery dead?  Or can Windows 7 have any option to not charge the battery?  It is a Dell Studio 15, I wonder if in such case, it looks like it can be a manufacture's defect, would Dell replace the battery? (if there was a recall, then they probably would, i think).  The notebook is 1.5 year old. Making an Emission shader emit a different colour of light than the colour assigned to the object? Executing .bat file In expressions such as "ham 'n eggs", the conjunction 'n appears to replace and, yet there is only one apostrophe to indicate the missing a and none for the missing d (i.e., no "ham 'n' eggs").    Is there a punctuation rule that governs this or is it just an idiosyncratic case? UMP of product functor What is this triangle icon [Samsung Galaxy S9+]? G-test vs Pearson's chi-squared test How to make a file (e.g. a .sh script) executable, so it can be run from a terminal