Say there's a bin of balls numbered 1 through 39, how many ways are there to pick 4 increasing numbers in a row?   First I figured that it would be a permutation without repetition problem, so I got that the number of ways of picking 4 different balls without considering increase is $$\frac{39!}{4!(39-4)!} = 82,250.999966$$  Then I thought I need to divide by some factor that changes the problem from any order to only increasing order which I thought might be 4!, so I got: $$\frac{82,250.999966}{4!} =  3,427.124998$$ and rounded down to get $3,427$. I'm terrible at these counting problems so I probably did this wrong. Can someone help me out please?
I have to figure out a way to count how many sequences are there s.t:   My alphabet is $1$ up to $49$. Each number that is chosen, is chosen only once. The sequence is $6$ digits long. The sequence is in increasing order.  e.g that is a valid sequence: $(1, 2, 40, 42, 43, 49)$  and this is not: $(1, 2, 40, 25, 25, 12)$   I know that for the first 3 rules the answer is: $\binom{49}{6}\cdot 6!$  How do I deal with the fourth?
Examples of patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."  Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".  I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.  So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:   be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.   I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers. What does standard deviation tell us in non-normal distribution In a normal distribution, the  imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? Which book (free or otherwise) was the most useful to you when you started learning LaTeX?  I am frequently asked this question by friends who want to learn LaTeX, and I recommend the book which got me started, , but I feel that there might be better options around.  Also see . Prove that $f$ is differentiable I created one event receiver and deployed it, but only users who are site collection admins are able to view this web part. I have included the SPSecurity.RunWithElevatedPrivileges but the same thing. What am I missing?  Below is my code   public override void ItemAdding(SPItemEventProperties properties)     base.ItemAdding(properties);     SPSecurity.RunWithElevatedPrivileges(delegate()        {SPWeb web = properties.OpenWeb();         base.ItemAdding(properties);         string name= web.Lists[properties.ListId].Fields["first name"].InternalName;          var list = web.Lists["location names"];    });} How do I know if a laptop will work with Ubuntu? I want to buy a laptop at some point in the near future, but not knowing beforehand whether it works with Ubuntu makes the whole operation risky and tiresome. I would like to know if there is some kind of official hardware database where I can enter the name of the laptop I'm interested in and see if it's 100% compatible out of the box, or in case it isn't what is it exactly that doesn't work (ideally also if there is a way to fix it).  Does Canonical have a project along these lines? Find path of standard executable for a given file type in Windows Suppose I am running cmd and I open an application by typing for example start document.pdf  I want to get the exact path of the executable that opens .pdf files  Of course I could search the entire system for the executable but I suppose it would be faster this way because the path is already stored somewhere  I don't know if it makes a difference but I am using Windows 7 Space between 2 paths in Illustrator Should questions where a problem arose from a typo be closed? Should questions where a problem arose from a typo be closed?  Here is a recent example:    The problem was a misspelled _gaq (written as _gat).   Should these be closed? Surely they're of no use to anyone once the OP recognizes the mistake. If so, what justification should be given? Not a real question? Too localized?     Just to be clear, the question linked above actually was not broken due to a typo. This was my mistake, and raises some other interesting questions. See the discussion below.  If anyone feels like editing this question with examples of actual typo questions, go for it. I am looking at some code and it has this statement:    ~ConnectionManager() {     Dispose(false); }   The class implements the IDisposable interface, but I do not know if that is part of that the tilde(~) is used for. Is spacetime discrete or continuous? (weak) homotopy equivalence What is projection of WGS84 in GIS software? WGS84 is a reference ellipsoid that determines coordinates of every point on Earth using latitude, longitude and height about the its surface.  This ellipsoid is often used as a base to make maps. Though, to make a map on a flat paper or a flat digital screen you need a projection too. An example of a popular projection would be UTM.   Mainstream GIS software give plethora of projection alternatives to display your data layers. Among them, they also let you display your data using WGS84 only.  I think they are using a certain projection but what projection is that? Finding places to play Magic the Gathering games A friend of mine recently got MTG for the XBox and is talking us into getting real cards.  I'm all up for starting but I dread the idea of playing the same two people over and over with no change epically since one of them is over competitive and exploits every loophole he can to win.  Does anyone know of any a reliable (not just google it) way to find places where people play MTG? I am using latest version of VirtualBox (VirtualBox 4.3.6 for Windows hosts) and after installing any OS and clicking inside of it mouse disappear. It is not visible until I press right CRTL. Anyone know what can cause this problem. I have also installed extension pack. It's the first time I'm using Tikz on Latex. I have to draw this block diagram (this is a low resolution snip from a paper)   In Tikz I have done this:   That is very similar but I need some adjustments, like the u_i arrow and the node sum after the H(s) block.  Is there anybody that can help me with this?  my code:      \tikzstyle{block} = [draw, fill=white, rectangle,  minimum height=2em, minimum width=3em] \tikzstyle{sum} = [draw, fill=white, circle, radius=1mm, node distance=1.5cm] \tikzstyle{input} = [coordinate] \tikzstyle{output} = [coordinate] \tikzstyle{pinstyle} = [pin edge={to-,thin,black}]  \begin{tikzpicture}[auto, node distance=2cm,&gt;=latex'] \node [input, name=input] {}; \node [input, above of=input](input1) {}; \node [sum, right of=input] (sum) {}; \node [block, right of=sum] (controller) {$K(s)$}; \node [sum, right of=controller] (sum1) {}; \node [block, right of=sum1] (filterinv) {$H^{-1}(s)$}; \node [block, right of=filterinv] (system) {$G(s)$}; \node [output, right of=system] (output) {}; \node [output, above of=output] (output1) {}; \node [block, below of=filterinv] (filter) {$H(s)$}; \node [block, above of=controller] (delay) {$D(s)$}; \node [sum, below of=sum1] (sum2) {};  \draw [-&gt;] (filterinv) -- node[name=u] {$u_{i}$} (system); \draw [draw,-&gt;] (input) -- node {$s_{i-1}$} (sum); \draw [-&gt;] (sum) -- node {$e_{i}$} (controller); \draw [-&gt;] (system) -- node [name=q] {$q_{i}$}(output); \draw [-&gt;] (q) |- (filter); \draw [-&gt;] (filter) -- node {} (sum2); \draw [-&gt;] (controller) -- node {} (sum1); \draw [-&gt;] (sum1) -- node[name=xi] {$\xi_{i}$} (filterinv); \node [input, below of=sum1](input2) {}; \draw [draw,-&gt;] (input2) -- node {$r_{i}$} (sum2); \draw [-&gt;] (sum2) -| node[pos=0.99] {$-$}      node [near end] {$\tilde{s}_i$} (sum); \draw [-&gt;] (delay) -| node[] {}      node [near end] {} (sum1); \draw [draw,-&gt;] (input1) -- node[pos=0.15] {$u_{i-1}$} (delay); \draw [-&gt;] (u) |- node[pos=0.99] {$u_i$}      node [near end] {} (output1);  \end{tikzpicture} Block network access of a process? Proof that Pi is constant (the same for all circles), without using limits How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? How to elegantly ignore some return values of a MATLAB function