Help me prove   $$\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}.$$
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}  $  Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer.   Good luck!  Here is what I got up to;  $\frac{(n+1)!}{(n-r)!(r+1)!} = \frac{(n)!}{(r)!(n-r)!} + \frac{(n)!}{(r+1)!(n-r-1)!}    $
What is the argument selector tracking a defined score (not [score_min=])? What if I want to make a command like this:  execute @e[score_A=10] ~ ~ ~ summon Creeper   So that the entity must have a score of 10 (not less, not more)?  Does it even exist that or a [score_A_max=] argument?  I know about scoreboard players test, but i guess it is not the case to use it. How can I remap a keyboard key? How to extend my root (/) partition? Ignore citations in captions in list of figures when numbering MySQL sort after argument in IN() Is it because Rey just saved BB-8 from a net, and then the droid just decides to follow her about - or is it for some other reason?  Does BB-8 have any idea who Rey is prior to being rescued by her?  Just wondering as she literally tells the droid to go elsewhere. Find the value of $\left[\frac{1}{\sqrt 2}+\frac{1}{ \sqrt 3}+......+\frac{1}{\sqrt {1000}}\right]$ Find the value of $$\left[\frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+.....+\frac{1}{\sqrt {1000}}\right]$$.Where [•] denote the greatest integer function.   I am very confused about this problem. I tried to find the upper and lower bound of the function. But I can't find any formulas to find the bounds. Somebody please help me. There is a lemma that says if a group $G$ has no proper nontrivial subgroups, then $G$ is cyclic. And here is the proof of the lemma:     Suppose $G$ has no proper nontrivial subgroups. Take an element $a$ in $G$ for which $a$ is not  equal to $e$. Consider the cyclic subgroup $\langle a \rangle$. This subgroup contains at least $e$ and $a$, so it is not trivial. But $G$ has no proper subgroups, so it must be that $\langle a \rangle = G$. Thus $G$ is cyclic, by definition of a cyclic group.   But here i do not understand the following: Why must $\langle a \rangle$ be a subgroup of $G$? For every single element $a$ in $G$, if $\langle a \rangle$ is a subgroup of $G$, then every group should have at least as many subgroups as the number of its elements. I would appreciate any help. Thanks When to use 'the' in front of a country? Java synchronized static methods: lock on object or class The Java documentation says:      It is not possible for two invocations of synchronized methods on the   same object to interleave.   What does this mean for a static method?  Since a static method has no associated object, will the synchronized keyword lock on the class, instead of the object? Why does flag marking as helpful/declined not always correlate with moderator action? The PS4 is now released in the EU, but due to shipment shortages is not available for sale unless pre-ordered a long time ago. Luckily, it seems as though Dualshock 4 controllers are availble for sale in a number of shops already.  I've long decided to buy the PS4 as well as a second Dualshock 4 controller, so I've been considering buying the "second" controller now in order to replace my old Xbox 360 controller, whose D-Pad is useless and whose RB button no longer responds correctly. But before I buy the Dualshock 4 early, I wanted to know how well my PC will support it.  I already know that , but what exactly does work, and what doesn't?  Can I expect the same key-mapping on the PC as on a PS4?  Basically, can I replace my Xbox 360 controller with a Dualshock 4 and still play games like Batman Arkham Asylum/City/Origins as if I were using the old Xbox 360 controller? recover data after using dd command I have a code that prints a table with 2 rows. I want to keep format of it and change 2 rows into 3.   \documentclass[conference]{IEEEtran}  \usepackage{pgfplots} \usepackage{siunitx} \usepackage{multirow} \begin{document}  \begin{table}[h] \caption{Family data set results} \label{tbl::Fam} \centering \begin{tabular}{|l|l|c|c|c!{\vrule width 2pt}l|l|c|c|c|} %\noalign{\hrule height 2pt} \hline \textbf{Column}&amp;\textbf{Algorithm}&amp;\textbf{min}&amp;\textbf{max}&amp;\textbf{std}l&amp;\textbf{Column}&amp;\textbf{Algorithm}&amp;\textbf{min}&amp;\textbf{max}&amp;\textbf{std}\\\hline \multirow{2}{*}{age}&amp;Original&amp;9&amp;2.8&amp;100&amp;\multirow{2}{*}{employer\_type}&amp;Original&amp;13&amp;2&amp;10\\ \cline{2-5} \cline{7-10}&amp;PrivBayes&amp;2&amp;95&amp;96&amp;&amp;PrivBayes&amp;2&amp;100&amp;100\\\hline \multirow{2}{*}{education\_level}&amp;Original&amp;7&amp;2&amp;98&amp;\multirow{2}{*}{marital\_status}&amp;Original&amp;10&amp;2&amp;100\\ \cline{2-5} \cline{7-10}&amp;PrivBayes&amp;2&amp;6&amp;96&amp;&amp;PrivBayes&amp;2&amp;6&amp;100\\\hline \hline %\noalign{\hrule height 2pt} \end{tabular} \end{table}  \end{document}   In this picture, I have 2 algorithms {Original, PrivBayes}. I want to change 2 algorithms into 3 algorithms with keeping the format of table. Can you please help me to do it? What cause the "from review" links to appear in comments? I sometimes see comments ending in "From Review" (hyperlinked to a review page) on questions and answers, e.g.        I spend a good bit of time in the review queues (on Unix &amp; Linux), but comments I make do not come with these "From Review" links. Note that I only have access to the "First post" and "Late answer" queues.  Do these links only appear for the higher-rep queues? Convert ArrayList to String[] array I have opened Disks program today and noticed that there is FAILING in the Assessment column for the SMART status.     What does this error mean? Is it serious problem?  I have Lenovo G50-45 laptop, disk is SEAGATE ST1000LM014-SSHD-8GB(LVD3) 1TB SSHD with 8GB NAND Flash (original disk in this latop).  Update 2017-09-24:  The disk has died yesterday. 1 year and 3 months have passed since 1st hint of problem. About 4 months ago I had few issues with it not booting, but I was able to make another clean install. Since yesterday the disk is non-responsive. As this was long expected I had backup of all important data.(!) The disk was about 2 years old, so that was quite premature death. Why exactly is work a path function? This might be a stupid question, but why exactly is work a path function?  In school, my teacher said it’s because it depends on the path. She gave an example of walking up a mountain, and said the work you expend depends on the path you take (straight path up or circling up around the mountain). But isn’t the work expended just equal to $E= mgh?$ Is it worth getting a suspension seat post as a cheap alternative to full suspension bike? I'm planning a ride from Birmingham to London on the Grand Union Canal (UK), which will be quite bumpy for most of the 145 mile distance.  I have a front suspension bike and can't afford a new full suspension bike.  Do suspension seat posts actually work?  If so, I'll get one of them.  Can anyone recommend a good model? What do I need in order to create an application for ubantu?