Evaluate the following sum using a combinatorial argument:  $$ \sum\limits_{k=0}^n {n \choose k} {m \choose k} $$  Can someone push me in the right direction with this? I thought for combinatorial proofs there has to be a left side and a right side where one side can be used to form a question? (if that makes sense? haha)  Is there a difference with combinatorial arguments? Any help would be greatly appreciated.
How do I evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.  I have tried to use binomial expansion and combine factorials, but I have gotten nowhere. I don't really know how to start this problem.  The answer is ${n+m \choose n}$. Any help is greatly appreciated.  EDIT: I'm looking for  a proof of this identity.
Map preserving intervals but discontinuous Is there a preinstalled or automatic firewall? Does Ubuntu come with a preinstalled or automatic firewall? If not, do I need one?   I've read some articles about the advantages of Linux over other operating systems about security (no need to have antivirus, ...) but would like to be sure. Converting Older United States Dollar Bills Slope of geometry as `fac` input for cycles material I have a terrain with two materials going into a Mix Shader, I want the fac input to be the slope of the terrain. Steep parts should be one color, flat part another. Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either? Example 4.21 in Baby Rudin: Why is the inverse of this function not continuous at this point? Here's Theorem 4.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:      Suppose $f$ is a continuous 1-1 mapping of a compact metric space $X$ onto a metric space $Y$. Then the inverse mapping $f^{-1}$ defined on $Y$ by $$ f^{-1}( f(x) ) = x \ \ \ \ \ \ \ ( x \in X ) $$ is a continuous mapping of $Y$ onto $X$.    Now in Example 4.21 Rudin illustrates that compactness of $X$ is essential in Theorem 4.17.   So here is Example 4.21:   Let $X$ be the half open interval $[0, 2 \pi )$ on the real line, and let $\mathbf{f}$ be the mapping of $X$ onto the circle $Y$ consisting of all points whose distance from the origin is $1$, given by $$ \mathbf{f}(t) = ( \cos t, \sin t ) \ \ \ \ \ \ ( 0 \leq t &lt; 2 \pi). $$ Then this function $\mathbf{f}$ is of course continuous and injective. But Rudin asserts that the inverse function is discontinuous at the point $(1, 0) = \mathbf{f}(0)$. How to demonstrate this fact rigorously?  So far in the book, Rudin has not discussed the functions $\cos$ and $\sin$. So can we come up with an example that uses only what Rudin has discussed? How to clear the BIOS security password on this notebook? I recently got an Acer Aspire E11 (E3-112m) and in order to install another OS I had to set a Supervisor password, so I used a well known password to me, the problem is that such password contains a special character ! which the password prompt won't let me input, though it let me when I had to set the password (buggy prompt I believe). Now everytime I need to change something in the BIOS I have to fail 3 times and go to  to generate a master key.  I have already tried several different methods to clear it:   Removing the CMOS battery Looking for the reset jumper (which doesn't even exist).   I also noticed there were 2 weird buttons on the front and back sides of the motherboard, I don't know what they are for but I also pressed them (while both the CMOS and the regular battery were disconnected). All of this without luck, the password just won't reset (though the RTC did).  Are there other ways to reset the Supervisor/Admin password on this model? For example i have the domainname buyhouse.com The keywords "buy house new york"  buyhouse.com/new-york/   buyhouse.com/buy-house-new-york/   Whats the best url option for SEO? Proving that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$ Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$ I am building a regression model and I need to calculate the below to check for correlations   Correlation between 2 Multi level categorical variables Correlation between a Multi level categorical variable and continuous variable  VIF(variance inflation factor) for a Multi level categorical variables   I believe its wrong to use Pearson correlation coefficient for the above scenarios because Pearson only works for 2 continuous variables.   Please answer the below questions   Which correlation coefficient works best for the above cases ?  VIF calculation only works for continuous data so what is the alternative?  What are the assumptions I need to check before I use the correlation coefficient you suggest?  How to implement them in SAS &amp; R? 19.10: Deactivate Alt Key From Triggering Applications' Menu Bar How can I stop the Alt key from triggering the menu bar of applications in Ubuntu 19.10?  I know that there are these , , and , as well as  but those solutions are no longer applicable. I did try searching for a similar option to the one in the answer of those questions but with no success. Why don't we pluralize "foot" in measurements? Problem on Extremal Graphs: three independent paths The problem states that:     A simple (or strict) graph with $n$ vertices and $m&gt;\dfrac{3(n-1)}{2}$ edges, has two vertex   joined by three independent paths.   Any hints, ideas or useful results, to attack this problem? Refreshing a question doesn't update the answer count properly In the Stack Exchange Android App v1.0.7 1.0.33 (with a HTC One running Android 4.4.2), the answer count for a question you're viewing is displayed in two places:   In the label of the answer tab:   At the bottom of the question tab:    I've noticed that these counts don't reliably update when new answers arrive.   If you refresh the question tab, neither count updates. If you refresh the answer tab, the second count will be updated, but the answer tab's count will not, leaving the two out of sync.   Refreshing either tab should probably update both counts!    Screenshots of the bug in action:    I waited for it to be answered. A while later, the question now has an answer.    The tab still says 0 answers though. Switching back to the question tab, , but the tab count still doesn't. Possible Duplicate:          When I add a piece of code of figure insertion to the context I'm faced with a problem: a number of irrelevant words appear.    Code:  \begin{figure}     \centering         \includegraphics[width=0.9\textwidth]{C:/Thesis/Latex/thesis_1(1)/Pictures/study area.jpg}         \rule{35em}{0.3pt}     \caption{The Grand St. Bernard wireless sensor network deployment (a) the coordinates of nodes according to the Swiss coordinate system (b) the distribution of the nodes in the study site \citep{r33}}     \label{fig:study area} \end{figure}   Problem: How can I create an AddIn to ArcGIS Pro and where can I find it from ArcGIS Pro application? What are these circular splotches on my negatives? Commas for multiple adjectives Why doesn't this perpetual motion machine using the buoyant force work? If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light? I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.  As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?  If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?  I quote Igor Ivanov (a stackexchange user from this question ) to give an insight into my question.     The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to  popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".   EDIT  After reading the answers I have a doubt.  SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy.