I'm taking a combinatorics course, and I was trying to practice from a book I took from the library. unfortunately there are no answers in this book, and there was a question I didn't know even how to approach to. I'd really appreciate it if anyone could give me an advice on how to think about it:     Given the group $\{1, 2, \ldots, n\}$,  choose $r$ different numbers so that no two are sequential. e.g for $r = 3$ and $n&gt;r$,  $\{1,3,5\}$ is a valid group but $\{1,4,5\}$ isn't valid.   Thank you very much for your time and attention!
In how many ways can we choose $r$ numbers from $\{1,2,3,...,n\}$,   In a way where we have no consecutive numbers in the set? (like $1,2$)
We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?  Furthermore, why can’t we call $ \infty + \infty $ and $ \infty \cdot \infty $ indeterminate forms?  I’m new to calculus, so please clear up my concepts if you can. Thanks! Should I upgrade my body or lens first? How can one change a coordinate system for a .lasd (Lidar) dataset?  The file is currently in UTM_Zone_15N and I am trying to convert it to IL_East. Can regular expressions be used to match nested patterns? Twice now I've tried pan frying fillets of white fish (Swai and Flouder) in olive oil and butter.  Both times the fish has disintegrated into unsightly mush.  Is there some technique I'm missing?  How can I pan fry without the fish falling apart? Why would a company care about the price of its own shares in the stock market? 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(Action and reaction)  But what intrigues me is that applying the same logic, the horizontal mass is being pulled to the right, it would have been understandable if the tension is directed to the left.   also Can I think of tension as the reaction force when you pull, and normal force when you push?  update If I was to think tension as a reaction force in the first picture for the horizontal mass, then can the frictional force be the action force? (Ehh, I don't think that makes sentence. Friction is always reaction) Mark questions to receive notifications and updates? Creating A Bootable USB Of Windows 8.1 On OS X? Reasons for wanting create such a usb include:   PC is unbootable and you want to prepare a bootable USB using the ISO file that you have on your external hard drive. to install Windows 8.1 on Mac without Boot Camp.   What are the necessary steps to create such a Windows 8.1 bootable usb on a Mac ? I need a poll inside my content types.  Sometimes the users create an article and like to create a poll directly into the node they created. They don't want to create an extra poll with no connection to the original article.  So how can I get the poll functionality inside a different content type? Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent?  Can we write down an explicit homotopy equivalence? It ought not to be too hard to get a map $\operatorname{cone}(f)\to \operatorname{cone}(g)$ from the existence of a homotopy $H:X\times I\to Y$, but I haven't been able to find a sensible candidate, despite trying for quite a while. How do I create variable variables? If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$) Let $A$ and $B$ be sets. Let $A\cup B\subseteq A\cap B.$ Prove that $A\subseteq  B.$   My understanding of this question is that all the elements of set $A$ that intersects with set $B$ exists in the union of sets $A$ and $B$ and because in order for the elements to intersect, the elements must exist in both sets $A$ and $B$. Therefore set $A$ is a subset of $B$.  My question is how do I write the proof down so that it is an acceptable answer? Also does this mean set $B$ is a subset of $A$? Is "The empty set is a subset of any set" a convention? I have been struggling for one hour with this simple problem (QGIS 1.8.0-Lisboa, Windows 7) :  I have a raster layer. I would like to display the label and not only the band value when I use the Identify Features button. Is it possible? Take the following C# class:  c1 {  event EventHandler someEvent; }   If there are a lot of subscriptions to c1's someEvent event and I want to clear them all, what is the best way to achieve this? Also consider that subscriptions to this event could be/are lambdas/anonymous delegates.  Currently my solution is to add a ResetSubscriptions() method to c1 that sets someEvent to null. I don't know if this has any unseen consequences. Does one wing of fly contain harmful microbes and the other contain only antibodies? There was  study but the conclusion and it's source's relability seemed a bit off. How come a house flies one wing contain all microbe and the other don't. How to create a progress bar? I want to delete all *.o files in a directory and its sub-directories. However, I get an error: sashoalm@aspire:~/.Workspace.OLD$ rm -r *.o rm: cannot remove `*.o': No such file or directory  On the other hand, rm *.o works, but it's not recursive.