Find the index on 1 in binary number of zeros
Find the highest order bit in C
Form inside a table Why is a nonstop flight sometimes more expensive than multiple connections? Usually I find a flight I like that uses one or more connections with the same or different companies. In these cases, if the distance between the first and the second connection is not too large, I prefer to go by bus or train and then take the flight. The problem is that sometimes it is more expensive to take the flight in the second airport than to take the flight from the first one. I mean, I'm wondering why sometimes it is more expensive to take only a flight than to take the same flight and another one.  A practical example (and real, you can check with SkyScanner). Imagine I am in Madrid (Spain) and would like to go to Rio de Janeiro (Brazil) on April 1st. The cheapest flight I have found is this:     TAM airlines: MAD -> GRU -> GIG : 507 €   Fantastic! GRU is one of São Paulo's airports. That's perfect for me! I'll go on to Rio traveling overland. But then, I look for this flight (MAD -> GRU, non-stop):     TAM airlines: MAD -> GRU : 597 €   It's the same flight, 90 € more expensive! I've found this kind of behavior a lot of times and I'm intrigued to know why it happens. Cant find my harddrives in ubuntu installation? Refactoring RadioGroup setter code? Syntax error while defining an array as a property of a class How to use the symbol \angle at the section title How do I find out which DOM element has the focus? I would like to find out, in JavaScript, which element currently has focus. I've been looking through the DOM and haven't found what I need, yet. Is there a way to do this, and how?  The reason I was looking for this:  I'm trying to make keys like the arrows and enter navigate through a table of input elements. Tab works now, but enter, and arrows do not by default it seems. I've got the key handling part set up but now I need to figure out how to move the focus over in the event handling functions. The successive difference of powers of integers leads to factorial of that power. Here's the formula:     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$   Can anyone give a proof of this result?  Note: The original question was to prove the more general     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$$   for an integer $l$, which some of the answers address. I am trying to mount my ext4 (sdc6) partition from a live usb because that's how I deleted two unused partitions left from a prior install of ubuntu, using GParted. The swap is mounted. However, I can't edit fstab, or find anywhere to click in nautilus to mount sdc6. In the terminal, I get the error: "can't find /dev/sdc6 in fstab". I'm afraid to exit the usb and update grub if that partition isn't mounted. Is there any easy way to do this, or do I have to install gksu or some other editor? Or do I need to mount it as sda6?  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *     2048 3905535 3903488  1.9G  b W95 FAT32  Device     Boot     Start        End   Sectors   Size Id Type /dev/sdc1            2048   29362175  29360128    14G 27 Hidden NTFS WinRE /dev/sdc2  *     29362176   29566975    204800   100M  7 HPFS/NTFS/exFAT /dev/sdc3        29566976  487023432 457456457 218.1G  7 HPFS/NTFS/exFAT /dev/sdc4       487024638 1250263039 763238402   364G  5 Extended /dev/sdc5       728393728  740290559  11896832   5.7G 82 Linux swap / Solaris /dev/sdc6       487024640  728391679 241367040 115.1G 83 Linux  Partition table entries are not in disk order. Evaluate   $$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$   $\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \left(\frac{(n+1)\cdot (n+2)\cdot (n+3)\cdots (n+n)}{(1)\cdot (2)\cdot (3)\cdots (n)}\right)$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \left\{\ln \left(\frac{n+1}{1}\right)+\ln \left(\frac{n+2}{2}\right)+\ln \left(\frac{n+3}{3}\right)+\cdots+\ln \left(\frac{n+n}{n}\right)\right\}$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{n+r}{r}\right) = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{1+\frac{r}{n}}{\frac{r}{n}}\right)$  Now Using Reinman Sum  $\displaystyle y = \frac{1}{2}\int_{0}^{1}\ln \left(\frac{x+1}{x}\right)dx = \frac{1}{2}\int_{0}^{1}\ln (x+1)dx-\frac{1}{2}\int_{0}^{1}\ln (x)dx =  \ln (2)$  My Question is , Is there is any method other then that like Striling Approximation OR Stolz–Cesàro theorem OR Ratio Test  If yes then please explain here  Thanks Prove that$$\overbrace{\binom n 0 + \binom n 1 + \binom n 2 + \cdots + \binom{n}{\frac{n-1} 2}} + \overbrace{\binom n {\frac{n+1} 2} + \cdots + \binom n {n-1} + \binom n n} = 2^n. $$   Background:  This problem came from the following link:    and I didn't understand why that was true in which no one on the page has proved it to be so. and i was wondering what was the proof on that I've found strange behavior when closing this  - I definitely see that there are 3 closing votes and 2 of them are at "Off-topic" section:   -but there are no explanations on section itself:   -while, normally, it's expected that it will appear. In custom comments section there are also no closing-votes displayed.   Perhaps it's just my browser (which is FF 26 x64 under Win) - so marking this as a "possible bug". Thank you for your attention. Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples? Proof of "Dual normed vector space is complete" This problem has been seen on mid-2012 MacBook Pros on Mavericks and on late-2014 Retina MacBook Pros on Yosemite.  One day while using iMessage on the Mac, all the contacts turned red and messages failed to send, immediately returning a "not delivered" error. It also reported when trying to resend the messages that the contacts were no longer associated with an iMessage account.  Things that didn't fix this behavior:   Deleting the com.apple.ids.madrid Removing the Apple ID information from my phone, turning off iMessage on both of the devices, and placing my iPhone in airplane mode for several minutes before turning it on logging in again.   Could someone help me figure out how to solve this? Simple past or present perfect? Which is preferable, simple past or present perfect, in situations like this one: I had a conversation with my friend and we planned to do something; shortly afterwards I call him back to announce the accomplishments. My biggest doubt about simple past is that perfect aspect is used to indicate past action (accomplishment) with current relevance, at the same time I'm not sure if I'm allowed to use present perfect, because there's a gap between accomplishing and the current moment, which is a characteristic of simple past. How do I check to see if a column exists in a SqlDataReader object?  In my data access layer, I have create a method that builds the same object for multiple stored procedures calls.  One of the stored procedures has an additional column that is not used by the other stored procedures.  I want to modified the method to accommodate for every scenario.  My application is written in C#. Do I need a Kuwait transit visa as an Indian national? I am an Indian national and passport holder, traveling from New Delhi to Frankfurt (via Kuwait) and back with Kuwait Airways. I see no Terminal change for the onward and return journeys. I have no plans to leave the airport due to short layovers.  Do I need a Kuwait transit visa? How do I preload a program for faster start? Is it possible to "preload"(actually not preload, but kinda emulate the startup) e.g. nautilus (or firefox) startup at login, so the first time I launch it myself it launches faster? Probably to start it on another desktop and close after some seconds? I had an (extremely minor) identity crisis when I clicked on my username and saw a different identicon than I'd had previously. :)  According to , your identicon is based on a hash of your last known IP address.  Since a user's profile picture part of their identity, I would prefer if my profile picture only changed through deliberate action on my part.  Since the user's IP hash is already saved (the identicon has to be displayed to people even during the unthinkable periods when we're not actively browsing), I propose preserving the first one assigned to a user, and not updating it even if the user's IP address changes.  Are there any upsides to the current behavior that I haven't thought of?