Attempting to Prove a lower bound for the Binomial Coefficient The inequality I am having a problem with is as follows:  Let $1\leq k\leq n$, where $n,k$ are natural numbers. Prove that $\binom{n}{k}\geq(\frac{n}{k})^k$  I have proven already the inequality $\binom{n}{k}\leq\frac{n^k}{k!}\leq(\frac{en}{k})^k$, but cannot seem to prove my lower bound. I have been told that differentiating may be of use, though I am unsure how to approach this. Any help is greatly appreciated.
Proving the inequality $\frac{n^k}{k^k}\leq\binom{n}{k}\leq\frac{n^k}{k!}$ involving binomial coefficient. What is the best way of proving inequalities including binomial coefficients in general? Lets look at this example:     $$\frac{n^k}{k^k}\leq\binom{n}{k}\leq\frac{n^k}{k!}$$   What I started with is:  $$\binom{n}{k}\frac{1}{n^k}\leq\frac{1}{k!}$$ $$\frac{n!}{k!(n-k)!}\frac{1}{n^k}\leq\frac{1}{k!}$$ $$\frac{(n-1)!}{(n-k)!n^{k-1}}\leq1$$ This is where I got stuck. How can I prove that is is lower than or equal to 1? Could you give me a hint?
How can I find color codes on Mac? I used to find color codes which I want to use for texts or background colors on htmls with color edit tool of MS paint which pre-installed on Windows. But now I use Mac instead of Windows. How can I find color codes with Mac? Is an software like MS paint pre-installed on Mac? How does the Mac choose which connection to use when both Wifi and Ethernet are connected? When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? I have an Ubuntu server and I've installed MySQL.  How would I use my two physical static IP addresses in my.cnf bind-address -- can I add two IP addresses? Why don't Death Eaters always use Avada Kedavra? When the Death Eaters attacked Hogwarts, why did the Death Eaters not just use Avada Kedavra and kill everyone? Here is the documents needed.     Here are the documents needed.       Which is grammatically correct and why? My guess would be the second one because of the plural form. I searched all over this site and the web for a good and simple example of autocomplete using jQuery and ASP.NET. I wanted to expose the data used by autocomplete with a webservice (and will probably do that next). In the meantime, I got this working, but it seems a little hacky...  In my page I have a text box:  &lt;input id="txtSearch" type="text" /&gt;   I am using jQuery autocomplete, set up per their example:  &lt;link rel="stylesheet" href="js/jquery.autocomplete.css" type="text/css" /&gt; &lt;script type="text/javascript" src="js/jquery.bgiframe.js"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="js/jquery.dimensions.pack.js"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="js/jquery.autocomplete.js"&gt;&lt;/script&gt;   Here is where it starts to get hacky... I call a page instead of a webservice:    &lt;script type="text/javascript"&gt;     $(document).ready(function(){         $("#txtSearch").autocomplete('autocompletetagdata.aspx');     });         &lt;/script&gt;   In the page I stripped out ALL of the html and just have this (otherwise, various HTML bits show up in the autocomplete dropdown):  &lt;%@ Page Language="C#" AutoEventWireup="true" CodeFile="autocompletetagdata.aspx.cs" Inherits="autocompletetagdata" %&gt;   And in my autocompletetagdata.aspx, I am using SubSonic to query, format and return data from the database (one data item per line):  protected void Page_Load(object sender, EventArgs e) {     // Note the query strings passed by jquery autocomplete:     //QueryString: {q=a&amp;limit=150&amp;timestamp=1227198175320}      LookupTagCollection tags = Select.AllColumnsFrom&lt;LookupTag&gt;()         .Top(Request.QueryString["limit"])         .Where(LookupTag.Columns.TagDescription).Like(Request.QueryString["q"] + "%")         .OrderAsc(LookupTag.Columns.TagDescription)         .ExecuteAsCollection&lt;LookupTagCollection&gt;();      StringBuilder sb = new StringBuilder();      foreach (LookupTag tag in tags)     {         sb.Append(tag.TagDescription).Append("\n");     }      Response.Write(sb.ToString()); }   If you don't do a LIKE query, then it returns everything that contains a match for the character(s) you type -- e.g., typing "a" will include "Ask" and "Answer" as well as "March" and "Mega." I just wanted it to do a starts with match.  Anyway, it works and it's pretty easy to set up, but is there a better way? How do I re-authorize Quicktime Player to use the microphone when once it has been denied? I wanted to use a USB audio input to record a video, and got something I've never seen before having done this 100 times; it asked "Authorize Quicktime to use the microphone?" I clicked no, because I didn't want it to use the internal mic, I wanted it to use a USB input.  Now, it won't let me get rid of the denial.  I've quit QT, quit &amp; reopened it twice and it still won't let me choose ANY input source. It still says it doesn't have access to the microphone.  Is this a new Mojave 10.14.2 bug or what?  How the heck do I get out of this?  Thanks! Gamma Distribution out of sum of exponential random variables I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function.  Well, I know that $P(T_i&gt;t)=e^{-\lambda t}$ and therefore $f_{T_i}(t)=\lambda e^{-\lambda t}$ so I need to find $P(T_1+\cdots+T_n&gt;t)$ and take the derivative. But I cannot expand the probability term, you have any ideas? Why doesn't an Environment texture appear when the world has a Volume Scatter shader? I'm working on lighting a scene in which I need to have both an Environment texture for the background and Volume Scatter for light cones, but when I check the volume section in World setting and choose the Volume Scatter, the environment becomes black background (but the lighting works as I want).  I'm using Blender 2.72  How can this be fixed? Python setup, install one module as a sub module of another module? Insert a full page image How can I insert a full page image?  No blank space left in the page.  Bonus:  How can I add some text over the image? Evaluation of $\int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$ Evaluation of $\displaystyle \int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$   $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{1}\frac{x\ln x}{\sqrt{1-x^2}}dx\;,$ Put $x=\cos \phi\;,$ Then $dx = -\sin \phi d\phi$  and Changing Limit, We get  $$\displaystyle I = -\int_{\frac{\pi}{2}}^{0}\cos \phi \cdot \ln(\cos \phi )d\phi = \int_{0}^{\frac{\pi}{2}} \ln(\cos \phi)\cdot \cos \phi d\phi$$  Now Using Integration by parts, We get  $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 \phi}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{(1-\cos^2 \phi)}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\sec \phi d\phi-\int_{0}^{\frac{\pi}{2}}\cos \phi d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-\left[\sin \phi\right]_{0}^{\frac{\pi}{2}}$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-1$$  Now How can I solve after that, Help Required, Thanks Is it possible to embed a scrollable frame inside a PDF? (This is an end-of-year crazy question)   Is it possible to embed a (scrollable) frame or a PDF inside a PDF? If so, how to do it in (pdf)LaTeX?  This is something pretty common in HTML (see for example ). Here it is a webpage with PDF frames   Here it is a fake screenshot so you get the idea.  Portable and non-portable (e.g. only-Acrobat) answers are welcomed.     Example of a down arrow below a frame. The up arrow can be on top of the frame and the triangles point up. I know there have been a lot of posts on this topic, but I haven't found exactly what I am looking for.   I noticed some dust spots on my sensor for my D750 after I took a picture with some sky in it and then was playing around with "dehaze" in Lightroom. Okay no problem, I have owned the camera for a year probably time to clean the sensor.   So I do a bit of research online and land on the fact that I can do this myself. I also want to learn how so I don't need to spend $50 or $75 each time I need this done. I decide to buy .  What I do is close my aperture down to F22, put my ISO on 100 and then set my shutter speed to 2 seconds or so and take a picture of a well lit white piece of paper.   I then go into Lightroom and jack the dehaze and look for dust spots. I see a ton. Okay, so now let's clean it. I wet the swab with a few drops to get it just saturated on the top of the swab, apply some pressure, wipe left to right, turn it over, wipe right to left and done.   Then I took the same picture and although the dust spots are different, they are still there! What the heck? Okay maybe it just needs a good blow out. I took my  and blow out everything I can inside of there, then I do the same cleaning method as the first time, except this time I start with a dry swab, throw it out, then get a new swab, wet it and clean it again.  I take the picture again... and they are STILL THERE. Okay let's get out the high power flashlight. I wouldn't say I scrubbed it, but I went back and forth with the swab on what seemed to be a stubborn spot (I know, sketchy). Then I clean it again with a normal swab and take another picture.   This went on and on and I have basically ran out of swabs. I still see spots. I used the same swab a few times (I know, also sketchy) because I just couldn't figure out why it wasn't getting clean. I even tried a can of air (although some don't recommend that because they have solvent in them, but I was desperate and some people on youtube did it). That actually seemed to help, but in the end I still have spots.  So obviously a lot of you will tell me to just "take it in! you're going to break your sensor", but I really want to learn how to do this. And now I am just stubborn.  The spots are at the top and bottom of the frame mostly.  Any ideas? I ordered  today, and I was going to try it out very carefully. I also ordered .       EDIT: I tried the gel stick. It did seem to help, but on the second try something else got on the sensor so I am going to wait until tomorrow to get the swabs and then try to do a swab cleaning + the gel stick again... we will see.  EDIT2: Well it looks . I think the gel sticky stick helped but it's still not spotless. I ran over it one more time. I also included a photo above to the people are nervous about cleaning their own sensor. Obviously I can't guarantee you won't screw up, but I ran a swab over this sensor at least 15 times and it still shots great. Can open, unsafe nuclear fusion reaction burn the atmosphere? I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts. Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature in "open air" would the atmosphere become a fuel for further fusion? Shouldn't the whole thing just be torn apart by its terrible pressure? Finite order endomorphisms Why the finite order endomorphisms are diagonalizable over the complex numbers (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with roots of unity on the diagonal? Can a single PS Plus subscription be used on two different profiles on the same PS4? Both my girlfriend and I play on the PS4 at home, so there are 2 profiles on PS4. When we set our profile in PS4, we set the console as the primary console.  Now we want to buy PSN Plus subscription, and I want to know if the same subscription can be used in both profiles, or  only in one.   In case the same subscription can be used in both profiles, how can we set it up? Or is automatic? I did not create a swap partition during Ubuntu installation.  Later, I freed up some space and made a swap partition.  Now after each boot up, I am manually opening gparted to right-click the swap partition to turn on the 'swapon' option.  How can I automatically enable the swap partition at boot?  Also, if I had not created the swap partition, what are my other options? I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ What is the meaning of the term "thread-safe"? Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment?