I need to prove that :  $$\binom{m+M}{n}= \sum_{i=0}^n \binom{m}{i}\binom{M}{n-i}$$  without expanding.  I started by seperating the left hand side to get:  $$\binom{m+M}{n} = \binom{m}{n}\binom{M}{n}$$, and am trying to open up the right hand side with Pascal's identity but I am not sure how to make it work out to the required equality.
A known identity of binomial coefficients is that $$ \sum_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}. $$ Is there a combinatorial proof/explanation of why it holds? Thanks.
Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? I got a visa with my mother when I was 17 years old. On it was stated "Not valid unless accompanied by my mother".    Can I travel to the UK alone now that I am 18? Suppose $f:X\rightarrow Y$ is a surjection and you want to show that there exists $g:Y\rightarrow X$ s.t. $f\circ g=\mathrm{id}_Y$. You need the AC to show this.  However, suppose $f$ is a injection and you want show there is $g$ s.t. $g\circ f=\mathrm{id}_X$. Then, according to my textbook, you don't need the AC to show this.  This is counterintuitive to me, because it's like you need a special axiom to claim that an infinite product of big sets is nonempty, while you don't need one to claim that an infinite product of singleton sets is nonempty, which seems smaller than the former.  So why don't you need the AC to show the latter?  EDIT: $X$ should be nonempty.  EDIT 2: I realized (after asking this) that my question mostly concerns whether the AC is needed to say that an infinite product of finite sets is nonempty, and why. SD Card showing weird file names and files are lost I just plugged my SD Card to my PC after a long videography day, and I suddenly got this problem:    It is also not showing all the videos I took on my camera so what is the solution? How does a new user begin using Stack Overflow/Stack Exchange? As a new user, I find it impossible to contribute to the community or to even begin contributing. I know why the reputation system is in place and I respect it, but at the same time it is too hard on new users.  It started with me wanting to upvote a question...     "Vote Up requires 15 reputation"   Just to appreciate a good question that helped me I need 15 rep? OK..., so I find out how I can get reputation points. More or less, I need to ask a question for +5 and add an answer for +10. I don't have any questions at the moment and I don't want to spam a random question, but I did happen to have a slightly different answer to the earlier question that I wanted to upvote. So I try to answer...     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.   Now... I know this doesn't apply to every question, but at the same time, this means I'm forced to answer yet another question, and I don't want to go around looking for random questions and spam irrelevant answers...  Now I'm stuck. So I try to talk to some other users in the chat rooms about how to go about getting reputation points and how they first started. I entered into a "casual chat room".     You must have 20 reputation on Meta Stack Overflow to talk here.   Yes... I can understand why this is enforced, because we don't want new users to spam chat rooms, but all I wanted to do was ask for help. So I can't even talk to anyone about my newbie problems. So okay, I'll just settle for upvoting comments in the question that helped me. I don't see an upvote hover-over button so I can't even do that.    So now I'm stuck at 1 reputation point, unable to cross that 15 rep barrier, and even after posting this question I'll only have 6 rep points and still unable to do anything, and I'll be forced to look for questions to answer. Yes, I should be contributing to the community, but I shouldn't need to begin answering or making questions to upvote questions and answers. This only spurs more spam/irrelevant answers if anything.   Very, very, very new-user unfriendly. Is it possible to make my laptop's screen act as a second monitor for my desktop computer? How to create a bootable Ubuntu USB flash drive from terminal? Is there any way to create a bootable Ubuntu USB flash drive from the terminal without using any third-party applications like , , ,  etc.  I tried to create a bootable Ubuntu flash drive with dd method,  sudo umount /dev/sdb sudo dd if=/path/to/ubuntu.iso of=/dev/sdb bs=1M   It creates files on the USB disk, but when I try to boot the USB disk it shows an Operating System Not Found error. In the Avengers movie, why is Hulk's behaviour inconsistent? In the Avengers movie, when      Bruce Banner is on the helicarrier, he becomes angry and transforms into the Hulk. Black Widow happens to be next to him and even though both are on same team, he attacks her and then Thor.  He becomes hostile to everyone on the ship.  But later on, in the fight against Chitauri troops, Hulk is very comfortably self-controlled.    Why is that?    The only thing I could think of is that it has been many days (a year?) since the Hulk had gotten angry, so he couldn't control himself.    But that's lame to be so puny when Bruce Banner was compared to Stephen Hawking in intelligence. Openlayers using Google Maps as base layer: possible to zoom out farther than max extent? See attached picture. This is how it is when the page loads. I'm currently using  map.zoomToMaxExtent();     Everything is working pretty well but I'd like to zoom out more/see the whole world using OSM/Google Map Layers: is this even possible?  Note that the zoom bar on the left is set to the lowest bar when the page loads (if you can't see the map well, just right click the image and paste into a new browser tab).   This  but even when I do MIN_ZOOM_LEVEL that doesn't do much.  Also note I am not setting any bounds, extents, or resolutions. Toying with those did not seem to do the trick either.   Any input is appreciated.   Edit: here's a codeshare if you want to take a look:   UPDATE:  This is not perfected yet (and it makes me think something may be wrong with my OL code itself) but currently I am using this   var extent = new OpenLayers.Bounds(-1.32,81.71,-1.18,-61.80).transform(new OpenLayers.Projection("EPSG:4326"), new OpenLayers.Projection("EPSG:3857"));   I tried   var extent = new OpenLayers.Bounds(-20037508.34279,-10018754.17139,20037508.34279,10018754.17139);   (both with and without transform) but this did not do anything.   UPDATE 2:   wrapDateLine: false works fine for now: can zoom out farther, pan around, etc etc.  Now I would like to cut off everything to the left of the date line on the left side of the map (the 'first' portion), and cut off everything to the right of the date line on the right side of the map (the '3rd' portion) WHILE still keeping the map full screen. I've tried bounds, resolutions, but no go. Is this even feasible?  Something like this but it should take up the full screen: possible to do? See below: Showing $\int_x^{x+1}f(t)\,dt \xrightarrow{x\to\infty}0$ for $f\in L^2 (\mathbb{R})$ Show that if $f\in L^2 (\mathbb{R})$ then    $$\lim_{x \rightarrow \infty} g(x)=0,$$   where    $$g(x)=\int_x^{x+1}f(t)\,dt$$   Since $f \in  L^2 (\mathbb{R})$ then $f \in L^2[x,x+1]$ for every $x$. Then  $$|g(x)| \leq \left(\int_x^{x+1} f(t)^2 dt\right)^{\frac{1}{2}} \int_x^{x+1} 1 dt$$ $$|g(x)|\leq \left(\int_x^{x+1} f(t)^2 dt\right)^{\frac{1}{2}}$$ Modified files in a git branch are spilling over into another branch Sadly, yesterday I've spilled some water on my MacBook Pro (El Capitan). Before you yell at me - it was closed, turned off and it wasn't a lot of water, the laptop was wet only on the top and under - not inside.  Today I've turned him on. At first battery wasn't recharging for some reason, but SMC reset fixed the problem.  At the first booting speaker was like talking to me, hard to explain but it was weird. Also only right speaker was "talking", despite the fact that only left side of the laptop was watered. Then there was no sound at all, not from the speakers, nor from the headphones. Also microphone's not working. USB ports are okay. The water was spilled from the left side of the laptop. I know this is hardware issue, and something's broken, but I want to be 100% sure. No it's not muted. I've also tried NVRAM reset a few times, but it doesn't fix the problem :(  There might be one factor, before spilling, my laptop "died" for the first time (battery energy ran off), but I doubt it's the case. (but hey, every info matters, right?)  It wouldn't be weird for me, if only left speaker wouldn't work, but built-in microphone not working is kinda weird in this case, isn't it?  I just want to ask you guys, what else can I check, before I'll be 100% sure it's dead inside?  I've plugged in external microphone - it's not working too. How to create a good glass reflection with Principled BSDF Converting spoken word audio files into text I've been working on transcribing a lecture series from audio to text files.  There are just over 20 lectures, and each one is about an hour long.  Using VLC Player, I'm able to slow down the speaker's words to a rate that I can keep up with.  However, it's extremely slow going, and it takes so much concentration that I can barely get through a third of a lecture before I have to take an extended break.  The thought had occurred to me that I would be better served by using some kind of speech recognition software, and then proofreading the computer-generated result.  This is likely to be a one-time thing, so while I don't mind using commercial software to get the job done, it would be preferable if it wasn't too expensive (or free; free is always good).  Command-line solutions are fine, too (:  Any suggestions? Setting Windows PowerShell environment variables If $f$ is a periodic function with $T = \pi$, does that mean that $F$ (a primitive of $f$) has the same period ? What about the inverse implication (periodic primitive implies periodic function) ? Can I use Beetel 450TC1 as wireless router? Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology. Show that every subgroup of $T$ is either dense in $T$ or finite.  How to prove the denseness? Which databases use real multidimensional indices? Is oracle ever using several indices to get data from tables, or will it always take the one that seems to have the highest selectivity? How about other dbms? It seems I put a   Bounty assigned by outside party, two lengthy, reference-citing answers, one "-1" (awarded the bounty), one "-2", two others scored "0" and "-2" respectively, the answers suggesting one or the other is correct, 73 comments and no consensus so far - and me, as the asker, lost without a clue what to think of the answers anymore -  this is no longer "Learners' Level" question, so I thought I'd bring it here and hear what the experts have to say.    Original question:  Plurality of verb depending on plurality of list elements  An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that - was my editor overzealous or am I trying to copy rules of my language that don't apply in English?    Someone linked a  for negatives, where the situation is more clear-cut ("There is no..."). Same applies to connection with "or" apparently, per accepted answer there. Still, nothing for "and".  It seems there is a consensus that in that if the verb goes after the list, it will be plural.     an orange, some grapes, two apples and a small pile of cherries were on the plate.   There doesn't seem to be clear consensus for:     On the plate { was | were } an orange, some grapes, two apples and a small pile of cherries   Adding more to the confusion is the abbreviated form: "There's"     One last note. In North America, at least, there is a widespread use of using a singular form like "there is" and "there was", without regard for the subject item or items, and this "there is" is often shortened to "there's":           There's three apples on the table!      Could you please clarify this mess? Allow [tag:tag-name] syntax to link to any site