In a race there are $n$ horses. In that race, more than one horse may get the same position. For example, a race with 2 horses can finish in 3 ways:   Both first Horse 1 first and horse 2 second Horse 2 first and horse 1 second   How can I find out the number of ways the race can finish for any $n$. Is there any recurrence relation to calculate the number of ways? I need help to understand the solving way with better explanation.
In a race there are n horses.In the race more than one horse may get the same position. For example, 2 horses can finish in 3 ways.   Both first horse1 first and horse2 second horse2 first and horse1 second   In how many ways,the race can finish so that a particular horse can never be first.If f(n) is the number of ways, is there any recurrence relation to get f(n).
I think that  is pretty good prima-facia evidence that  is just facilitating the deposition of crap. Stack Exchange doesn't know when I last checked my global inbox? I am subscribed to get my unread inbox items every 3 hours. It's working well, but today I noticed something quite weird:    For sake of future search, adding the text:  The following item was added to your Stack Exchange global inbox since you last checked it on 2015-05-18  But that's just not true, I checked my inbox yesterday. Not 7 months ago. Can it be fixed, or did I miss something? Even going to my  did not &quot;fix&quot; this, got today another email with the same date as yesterday. In HP-UX server, grep -r option is not working In unix server grep -r option is not working. OS is HP-UX.  If grep -r is not working any alternative way to do recursive search without -r option? Collection was modified; enumeration operation may not execute I can't get to the bottom of this error, because when the debugger is attached, it does not seem to occur.  Collection was modified; enumeration operation may not execute  Below is the code. This is a WCF server in a Windows service. The method NotifySubscribers() is called by the service whenever there is a data event (at random intervals, but not very often - about 800 times per day). When a Windows Forms client subscribes, the subscriber ID is added to the subscribers dictionary, and when the client unsubscribes, it is deleted from the dictionary. The error happens when (or after) a client unsubscribes. It appears that the next time the NotifySubscribers() method is called, the foreach() loop fails with the error in the subject line. The method writes the error into the application log as shown in the code below. When a debugger is attached and a client unsubscribes, the code executes fine. Do you see a problem with this code? Do I need to make the dictionary thread-safe? [ServiceBehavior(InstanceContextMode=InstanceContextMode.Single)] public class SubscriptionServer : ISubscriptionServer {     private static IDictionary&lt;Guid, Subscriber&gt; subscribers;      public SubscriptionServer()     {                     subscribers = new Dictionary&lt;Guid, Subscriber&gt;();     }      public void NotifySubscribers(DataRecord sr)     {         foreach(Subscriber s in subscribers.Values)         {             try             {                 s.Callback.SignalData(sr);             }             catch (Exception e)             {                 DCS.WriteToApplicationLog(e.Message,                    System.Diagnostics.EventLogEntryType.Error);                  UnsubscribeEvent(s.ClientId);             }         }     }          public Guid SubscribeEvent(string clientDescription)     {         Subscriber subscriber = new Subscriber();         subscriber.Callback = OperationContext.Current.                 GetCallbackChannel&lt;IDCSCallback&gt;();          subscribers.Add(subscriber.ClientId, subscriber);                  return subscriber.ClientId;     }      public void UnsubscribeEvent(Guid clientId)     {         try         {             subscribers.Remove(clientId);         }         catch(Exception e)         {             System.Diagnostics.Debug.WriteLine(&quot;Unsubscribe Error &quot; +                      e.Message);         }     } } Set margins in maketitle Is it possible to change the margins in make title, so they match the margins of the document?  I have created the following modification to the \maketitle. I would like everything to align to at the same left margin.    \documentclass[a4paper]{article} %=============================================== % Title and author \makeatletter \def\@maketitle{%   \newpage   \vspace*{-\topskip}      % remove the initial space    \flushleft Incomplete draft. Please do not cite or quote without premission   \vskip 2em   {}   \begingroup\flushleft    % instead of \begin{center}   \let \footnote \thanks   \hrule height \z@        % to avoid the insertion of lineskip glue     {\LARGE \protect\textbf{\@title} \par}%     \vskip 1.5em     {\large       \lineskip .5em          \@author         \par}%     \vskip 1em     {\normalfont       Last updated on \@date}   \par\endgroup            % instead of \end{center}   \vskip 1.5em             % &lt;--- modify this to adjust the separation } \makeatother %============================================== \title{Physician Information Acquisition In a Dynamic Setting} %\email{rud} \author{ \large {\Large Rud Faden} \\ \textit{\normalfont University of Copenhagen} \\ \textit{{\small \href{mailto:rudfaden@gmail.com}{rudfaden@gmail.com}}} } \date{\today} % no date \usepackage{hyperref} \begin{document} \maketitle \abstract{Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enimad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui ocia deserunt mollit anim id est laborum.}    \section{First} Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. \end{document} If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$. Let $a,b,c$ be the lengths of the sides of a triangle. Prove that $$\sum_{\text{cyc}}\frac{a}{b+c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}&lt;2\,.$$  Attempt.  By clearing the denominators, the required inequality is equivalent to $$a^2(b+c)+b^2(c+a)+c^2(a+b)&gt;a^3+b^3+c^3\,.$$ Since $b+c&gt;a$, $c+a&gt;b$, and $a+b&gt;c$, the inequality above is true.  Is there a better, non-bruteforce way? Recommended shutter speed for action sports? What shutter speed do professional sports photographers typically use for action sports like football and basketball? Careers ad asking if I want to work where I already work? Short Story about interminable rain on Venus It's not All Summer in a Day.  This was a story about explorers on Venus, who are separated from their base (possibly they're lost, possibly they suffered equipment failure, possibly they're just traveling from base to base) and are slowly being driven insane because it never stops raining.  The story is old enough that Venus' clouds were still believed to indicate a wet climate.  I can't remember much more detail than that.  I have the impression that the narrator at one point either dreamed of or fantasized about walking into the destination base and being handed a very plush towel to dry off.  I can't remember the central plot points - only the (very wet) setting.  I would have read this in the late 1970's or early 1980's.  I thought it might be an H. Beam Piper short story but I've been through his catalog and don't see any likely suspects. Is there any way to hide taskbar labels and not combine taskbar buttons? There are three options for customizing taskbar buttons in Windows 7:   Always combine, hide labels Combine when taskbar is full Never combine   Is there any way to hide labels and not combine taskbar buttons? Or even better, to exclude only certain applications from combining?  My main problem is that I switch a lot between two instances of a single application, and having to hover over combined icon and then choosing the correct one. Alt-Tabbing is not a good alternative when I'm switching between more than three applications; clicking on taskbar only once is what I'm looking for. Serana disappeared during dawnguard questline. Choosing my name as an author when publishing a scientific paper, can I use my "unofficial" first name? I have one official first name which appears on my ID card. I have another first name which friends call me by. Is it possible to use my unofficial first name and official family name to publish scientific papers? How can I prove to someone that I am one of the authors? Linear mixed effect model interpretation with log transformed dependent variable I have a dataset in which the response variable is non normal, but on log transforming, it follows the normal distribution. I constructed a mixed effect model using lme4::lmer() as below (multiple measurements within each region), where A is a continuous variable and there are 7 regions  model&lt;-lmer(log(response)~A + (1|Region), data, REML=FALSE)   My output is   Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's method ['lmerModLmerTest'] Formula: logresponse ~ A + (1 | Region) Data: df[, -c(6)]   AIC      BIC   logLik deviance df.resid  151.8    159.8    -71.9    143.8       51   Scaled residuals:   Min       1Q   Median       3Q      Max  -2.24856 -0.63631  0.02206  0.59999  1.94895   Random effects: Groups   Name        Variance Std.Dev. Region   (Intercept) 0.8611   0.9280   Residual             0.5814   0.7625   Number of obs: 55, groups:  Region, 7  Fixed effects:               Estimate Std. Error        df t value Pr(&gt;|t|)   (Intercept) -1.112047   0.747373 19.651457  -1.488   0.1526    A           0.008585   0.004250 29.157266   2.020   0.0526 .  Correlation of Fixed Effects: (Intr) A -0.871   1) Would I infer that for A effects the response variable and the response increases by exp(0.008585) +/- exp(0.004250) (std error)? Would that be the right interpretation?  2) exp(Intercept) is not equal to the mean(log response). They are close but not equal. Why are they not equal and should I be concerned?   I know that this is a hard question to answer without knowing the dataset. So please suggest how I can improve the question. How should I phrase an important question that I need to ask a professor? When needing to ask a professor about something important and potentially delicate, what is an appropriate way to phrase the question or write the email?  Examples include:   Requesting a recommendation letter Asking for exceptions to policy (e.g., ignoring a course prerequisite, rescheduling an exam) Asking if you can do research with them, join a group, get into a program Scheduling a meeting, exam, or defense Letting them know about a change of plans Letting them know about a life event affecting your work or studies (e.g., wedding, funeral, sickness, vacation) Why the switch statement cannot be applied on strings? Compiling the following code and got the error of type illegal.  int main() {     // Compilation error - switch expression of type illegal     switch(std::string("raj"))     {     case"sda":     } }   You cannot use string in either switch or case. Why? Is there any solution that works nicely to support logic similar to switch on strings? What setting causes OS X to let you resize windows behind the dock? I've set up my fair share of Macs over the years, but I've never been able to pinpoint what setting tells OS X that you're ok with being able to drag-resize a window vertically so that it goes behind the dock (assuming the dock is in its default position at the bottom of the screen), as opposed to only being able to resize it vertically so that it is touching the top of the dock's "space".  My work computer is a 21" iMac Core 2 Duo and it drives me crazy because I'll quickly drag-resize a window larger by pulling down and to the right but instead of staying locked vertically above the dock it'll slide behind it which I don't like so then I've gotta jiggle the window around to get it all back into visible space... my home computer (6 core tower) doesn't have this problem even though I've, from what I can tell, set it up nearly identically.  Both machines running the latest version of snow leopard.  Here's a video captured from my machine at home showing the window resize behaviour I prefer. If it helps at all I can do another one from my work machine tomorrow showing the windows resizing behind the dock. Center of a free product Getting polygon boundaries of City in JSON from Google Maps API? When searching Google Maps for a city, the city's administrative boundary is displayed.    Is there a way to get the polygon boundaries of City in a JSON format using google maps api? With biblatex, alphabetic style, I have     &lt;author&gt;. &lt;title&gt;   But I want to have a colon ":" instead of the period "." after the author:     &lt;author&gt;: &lt;title&gt;   What do I have to change?