Find the value of $$\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}\frac{k-i}{M-100},$$ where $M-k&gt;100,k&gt;100$.   My approach: The Vandermonde's identity states that $$\sum_{i=0}^r\dbinom{m}{i}\dbinom{n}{r-i}=\dbinom{m+n}{r}$$ for any non-negative integer $m,n,r$.   Therefore setting $m=k$, $n=M-k$ and $r=100$, in the Vandermonde's identity we have, $$\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}=\dbinom{M}{100}.$$  This implies that $$\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}\frac{k-i}{M-100}$$ $$=\frac{k}{M-100}\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}-\frac{1}{M-100}\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}i$$$$=\frac{k}{M-100}\dbinom{M}{100}-\frac{1}{M-100}\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}i.$$  I need help in evaluating the expression $$\sum_{i=0}^{100}\dbinom{k}{i}\dbinom{M-k}{100-i}i.$$
Show that    $$\frac{\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}}{\binom{M}{n}}=\frac{k}{M},$$ where $M-k&gt;n$ and $k&gt;n$   From Vandermonde's identity, I get $\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}=\binom{M}{n}$.   But what about the sum of the product in the numerator? Any hint will be helpful.
Which of the following is a correct sentence:   Does anyone know where's Linda?  Does anyone know where Linda is?    Does anyone know why and how to use above usage if #2 is correct. Also please let me know this kind of specific structures. Explaining Infinite Sets and The Fault in Our Stars In watching The Fault in Our Stars I could not help but cringe at a line that flew in the face of mathematics and subsequently ruined the movie for me:     "There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities." - John Green   While walking out of the theater I tried to explain to my friends why there were, in fact, exactly the same amount of numbers between 0 and 1 as 0 and 2, but Cantor and bijective functions are not great learning tools to English majors.  Does anybody have an eloquent or elegant way to enumerate this phenomenon using an example accessible to those not familiar with advanced mathematics? Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$ I am trying to prove the above with the conditions that N is normal to G and $G/N$ is abelian.  So then for $Na,Nb\in{G/N}, NaNb=NbNa$.  But $NaNb=Nab=Nba$, so then we know $ab=ba$, so G is abelian.  So if $ab\in{G}, N=abN(ab)^{-1}=abN(ba)^{-1}=baNa^{-1}b^{-1}....$ and i'm getting lost... \varepsilon vs. \epsilon React on global hotkey in a Java program on Windows/Linux/Mac? A Java6 application sits in the system tray. It needs to be activated using a hotkey (e.g. Super-G or Ctrl-Shift-L etc) and do something (e.g. showing an input box).  How do I do that on:   Windows (XP or Vista) OS/X Linux (Gnome or KDE) How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? Prove that $\lim_{n \to \infty} a^{\frac{1}{n}} = 1$ if $a &gt;0$.  In my textbook, we are given a suggestion to let $a^{\frac{1}{n}} = (1+h_n)$ and then show that the $h_n$ term goes to zero using a Theorem that states the following conditions:  $\lim_{n \to \infty} = \infty,$ if $a &gt;1, 1,$ if $a = 1,$ and 0, if $\vert a \vert &lt; 1$. I apologize for not formatting the above cases appropriately; I could not figure out how to use a giant left brace to group them altogether.  If I rewrite it as $a^{n^{-1}}$, then I thought that this might help, but I don't think the Binomial Theorem would then apply since the exponent is negative. Furthermore, the problem is only concerned with the positive, real $n$-th roots. I am quite stuck on this problem and any suggestions or advice would be greatly appreciated.   I am using the textbook Introduction to Analysis by Arthur Mattuck. Using a foreign driver's license in California when staying with a J-1 visa E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so? Why is posterior density proportional to prior density times likelihood function? According to Bayes' theorem, $P(y|\theta)P(\theta) = P(\theta|y)P(y)$. But according to my econometric text, it says that $P(\theta|y) \propto P(y|\theta)P(\theta)$. Why is it like this? I don't get why $P(y)$ is ignored. Why should users never use normal sudo to start graphical applications? I've read  and am interested in this line:     You should never use normal sudo to start graphical applications as Root.   Why? What is the difference? Please provide a simple explanation, as I'm just a normal desktop user. Dimensioning of a technical drawing in TikZ Out of five needed members for my doctoral thesis committee, four are fixed. For the fifth member, I approached Mr.A. He has no real connection to my research or topic but I contacted him because he was just as good a fit as anybody else in the department to be the fifth member. I have never worked with him or taken any classes but we did talk a lot in the first few years where he would just ask about my progress by the copy machine, etc. We don't even have a "rapport". The relationship is absolutely neutral, nothing positive/friendly, and nothing negative. I asked him to be on my committee and he said yes. I said I'll get back to you with dates, etc. He is a senior tenured professor.  Then about a month later, I was looking at the CV of some of the new hired (junior) faculty in the department Mr.B (this is his first year) and I see that we have a collaborator in common. Mr.B has published multiple papers with him and Mr.B is working with the same code and the code managing team that we are working with. He is definitely a good person to have on my committee. I finally see Mr.B after a few more months (because of busy schedule/travel/etc.) and he agrees to be on my committee.  The question is, how can I tell Mr.A, as tactfully as possible, after about four months of being quiet that he is basically rejected from my committee because I didn't know about Mr.B but now that I know, Mr.B is a much better match than he is? That I don't need him but I do appreciate his offer? I don't want to antagonize Mr.A or burn any bridges. Should this be in email or in person? Should I explain or just keep it terse? Should I keep it vague that I don't need him and thanks for his time? Or should I tell him about Mr.B and why he is a better match?  FYI, this was before anything official/paperwork was done so there is no problem with that. How to fix pagination for custom loops? Looking for a sci-fi movie in which a human astronaut befriends an alien on a reddish planet This probably wasn't a blockbuster back in its time as I remember the special effects didn't seem all that bombastic, but I do remember a few details.  A human astronaut lands (or crash-lands) on a prominently reddish planet. It might have been Mars, thought it might not have been. I think the atmosphere on said planet was humanly breathable (as in the astronaut does not wear his helmet).  This human astronaut needs to survive. He spots a humanoid alien, apparently also having crash-landed or otherwise in need of survival. He sees this alien consume some sort of greenish orb, apparently for nourishment. He sneaks up to the alien (or the alien craft) in an attempt to grab some of those greenish orbs. I don't remember whether he succeeds, but he does get beaten up by the alien.  The next bit of detail I remember was the human astronaut constructing some sort of shelter to protect both him and the alien from... meteor showers? The human astronaut piles up some rocks to build a hut while the alien, apparently having learned a few English words, says "shiiiiit" and "no solid". The human astronaut asks, incredulous, "No solid?" and punches the hut he just build. "See? Solid!" he proclaims, while the hut collapses a few seconds later. The alien laughs.  Later on the human astronaut is either caught by or accidentally fell into some sort of mouth-in-the-sand creature. He is saved by the alien in the process (or was it the alien who fell into the mouth-creature and was saved by the human astronaut). They witness the mouth-creature spitting out a turtle-like shell (from another local wildlife) of some sort. The human astronaut observes, and I paraphrase, "This stuff must be so tough that even [the mouth-creature] cannot digest it!" He then proceeds to build a meteor shelter from collected shells.  The last bit of detail I remember is when the alien becomes pregnant and is dying. The human astronaut, having thoroughly befriended the alien at this point, asks (and I paraphrase), "What do you mean you're pregnant? Aren't you a male?" to which the alien responds (and again I paraphrase), "For us it comes naturally, unlike for you humans whose pregnancy only comes after a brief period of pleasure". The alien then asks the human astronaut to bring the child to the alien's home planet and recite the names of the child's ancestors in front of... alien rulers, I guess. The last scene of the movie is of the human astronaut, having arrived at the alien's home planet, standing side-by-side (or perhaps hand-in-hand) with the alien child while chanting in an alien language (what I presume to be the names of the child's ancestors). Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$ Got problem with installation. "No root file system is defined." I'm very new with Ubuntu and generally with linux. I saw ubuntu and it seems like this OS is really good and stable, and so I decided to install it alongside my windows 7 OS.  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Here is some more info:    Here is the RAM I'm looking at:    It all looks compatible to me, I just was curious about the four channel thing. Is "funnest" a word? We're planning to upgrade an existing farm from 2007 to 2013. I learned that the best way and maybe the only way to achieve this is by migrating in two steps. 2007 to 2010 and 2010 to 2013. The actual farm is a sql-server(2008r2 64 bit) and a sperate sharepoint-server(web-frontend, etc....) I want to keep the sql-server as it is, if possible and we want to install a new 64bit host for sharepoint 2013. We will design a new branding for 2013 afterwards, so we need to keep the content, but not the layout. There's so much about migration available on the web, but Can I find something like best practises? The result of ls * , ls ** and ls *** I know using the command ls will list all the directories. But what does the ls * command do ? I used it and it just lists the directories. Does the star in front of ls mean how deep it will list the directories?