Reading numbers
How to do binary calculation？
$k$ times's draw from $n$ numbers with replacement, each number at least appear once Sample $k$ times with order from $n$ distinct numbers with replacement where $k\ge n$. Here "with order" means the sample is treated like a sequence rather than a set, so results like "1,3,3" and "3,1,3" are different.  The question is what is the probability that every number is sampled at least once? My answer is $1-\frac{{\sum\limits_{i = 1}^n {C_n^i\times i! \times {{(n - i)}^k}} }}{{{n^k}}}$ where $C_n^i$ denotes $n$ chooses $i$.  ${C_n^i\times i! \times {{(n - i)}^k}}$ means first choose $i$ ($i=1,2,...,n-1$) of numbers not to appear in the sample, and then we sample from the remaining $n-i$ numbers for $k$ times.   My question is, is there more elegant solution where the numerator can be directly computed? i.e. can we directly calculate the number of possible sample sequences in which every number has at least one occurrence? Thank you!    PS: my attempt was $C_k^n\times n!\times n^{n-k}$, meaning that we first choose $n$ positions in the sequence of length $k$ to place $n$ numbers, and then permute the $n$ numbers. This is to ensure every number to appear once. The remaining $n-k$ are sampled from the $n$ numbers without restriction.   However, I later found this seems not correct, and it counts the same sequence multiple times.  PS2: as pointed out in the comments, my first answer "1- ..." is not correct either. Getting around the FAT32 4GB file size limit I've got a repo that I contribute to within an Organization that I administer. I'm trying to add it to careers, but it won't allow it. Is this on purpose? I don't really want to fork it to my personal account, since I won't be maintaining it from there. Entire function with vanishing derivatives? Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes.  What can you say about $f$?  A hint suggests that $f$ must be a polynomial. Which form should be used for at­tribu­tive nouns like “stu­dent union”: sin­gu­lar or plu­ral, or pos­ses­sive sin­gu­lar or pos­ses­sive plu­ral? When should a noun that’s used at­tribu­tively to de­scribe an­other  noun be plu­ral, and when should it be sin­gu­lar?  And when should it be pos­ses­sive, like bak­er’s dozen and when should it be plu­ral pos­ses­sive, like farm­ers’ mar­ket?  In other words, why do we say teach­ers union rather than teacher  union?  And why do we say wed­ding plan­ner rather than wed­dings plan­ner?  Which of these vari­ants is or are cor­rect?        stu­dent union   stu­dents union   stu­den­t’s union   stu­dents’ union      What about com­mu­nity val­ues ver­sus com­mu­ni­ty’s val­ues?  Please note I am look­ing for a gen­eral rule or at least some tips.  These are only ex­am­ples.  Up­date: It seems even na­tive speak­ers fol­low their per­sonal  style to write such com­pound words. I won­der why in the IELTS lis­ten­ing sec­tion, the lan­guage learner has to write a spe­ci­fic form and there is no rule for this. How to generate a simple cartesian plane system? I need to make a cartesian plane for plotting the straight line joining the points (-3,1) and (6,-2). This is a very simple cartesian plane plotting. When I looked at the manual for tikz the graphics for cartesian planes are for more advance plotting, and what I need is very simple.  Something like this: There are many addons. Do you know how show it in two cloumns? Geometric proof that a Pythagorean triple has a number divisible by $3$? Can I infer from “If $a,b \in \mathbb{N}$ such that $(a,3)=(b,3)=1$ then $a^2+b^2\neq c^2$ for any $c\in\mathbb{N}$” that any pythagorean triple has a number which is divisible by $3$? If so, how can this relationship be proved geometrically? Why multiple capacitors in parallel? Wondering why you would choose to use smaller capacitors in parallel vs one bigger one.  Here's an example:      Is this a component size and track width choice?  Thanks! Prove that 10101...10101 is NOT a prime. So basically we have a number $10101...10101$ that contains $2016$ zeros and can be written as$ \sum _{ k=0 }^{ 2016 }{ 100^{ k } }$ . I want to prove that this number is not a prime without using anything besides a piece of paper and a pen. I'm stuck on this for quite a few days now. Get PHP to stop replacing '.' characters in $_GET or $_POST arrays? If I pass PHP variables with . in their names via $_GET PHP auto-replaces them with _ characters. For example:  &lt;?php echo "url is ".$_SERVER['REQUEST_URI']."&lt;p&gt;"; echo "x.y is ".$_GET['x.y'].".&lt;p&gt;"; echo "x_y is ".$_GET['x_y'].".&lt;p&gt;";   ... outputs the following:  url is /SpShipTool/php/testGetUrl.php?x.y=a.b x.y is . x_y is a.b.   ... my question is this: is there any way I can get this to stop? Cannot for the life of me figure out what I've done to deserve this  PHP version I'm running with is 5.2.4-2ubuntu5.3. Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that there exists a simple closed curve, which is not null-homotopic in $U$.  I am sure that this is a standard problem, and I am looking for an elegant, if possible, solution or a reference. I have in mind the following sketch of proof:  a. Construct a polygonal closed curve $\tilde\gamma$ which is homotopic with $\gamma$ in $U$.  b. Construct $\tilde\gamma$ so that it intersects itself in finitely many points, and not whole segments.  c. Use induction on the number of self–intersections of $\tilde\gamma$, in order to show that $\tilde\gamma$ is written as a product (in the fundamental group's sense) of simple closed polygonal curves. Order discrete x scale by frequency/value OK...before everyone blasts this with references to the relativistic invariance of the physical laws, time dilation, etc let me add some context. Also, I am an amateur with an interest in physics, so I don't know the details of the physics. However, I have read enough about the cosmological principle and cosmic time to develop this "confusion" .  Based on evidence gathered to-date, it appears that the universe is both , as evidenced by observations and the . This principle, when combined with the FLRW metric, allows space-time to be divided into non-intersecting slices and therefore establish a "cosmic time."   Now, the existance of such a time in no way suggests that it is the "right" time or a true "now", hence violating relativity -- although some presentist philosophers (e.g., W.L. Craig) have tried to make this argument, but it is not generally accepted in philosophical circles.  However, this leaves me confused about an apparent disconnect between relativity and the cosmological principle:  Isotropy only holds if we are at rest relative to the cosmic microwave background radiation (universal rest frame), o/w anisotropy is present - yet each reference frame is supposed to allow equally valid observations. How can these two be reconciled when most reference frames would lead us to conclude that the universe is NOT isotropic?  It seems that only by appealing to the idea of being at rest relative to "universal rest frame" can we explain away any discrepancies from isotropy as due to "peculiar motion". However, doesn't this give this "universal rest frame" and its associated time an empirically privileged status, even though physical laws work just fine in every reference frame?   For example (pardon any abuse of astronomy): if look off in some part of space and see only quasars, and then in another part of space and see only brown dwarfs, but I measure both as apparently the same distance from me, then can I conclude that we are in motion, since otherwise we would have an empirical contradiction (kind of like finding dinosaur and human remains in the same strata)?  Any help on where I am going wrong would be helpful. Intuitively, I don't think there should be a way to establish absolute ordering between space-like separated events, but the above reasoning suggests otherwise. I'm writing my final report in LaTeX for my bachelor's degree. I'm aiming at having a document similar to the .doc template used normally.   I want to have my customs headers and footers applied on pages when I'm starting a new chapter.  I'm using documentclass report with fancyhdr and lastpage packages.  My headers are defined like this:  \lhead{\myTitle} \rhead{\today} \cfoot{\thepage\ of \pageref{LastPage}}   My chapters are created using standard \chapter{title} macro.  On pages with new chapters, the only headers/footers I see is the standard page number at the bottom.  What should I do?  EDIT: I've finally gone with Peter Breitfeld's solution. I'Ve gone a step further and created my own chapter command:  \newcommand{\myChapter}[1]{     \chapter{#1}     \thispagestyle{fancy} }   Thanks for the help! How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm wondering if there's a simpler way to computing all the possible permutations, as an alternative to manually calculating all the possible 6-letter combinations and summing the corresponding permutations of each. In addition, is there a method to generalize the result based on any P number of letters in a set with repeating letters, to calculate the number of n-letter permutations?  Sorry if this question (or similar) has been asked before, but if it has, please link me to the relevant page. I also apologize if the explanation is unclear, or if I've incorrectly used any terminology (I'm only a high-school student.) If so, please comment. :-) In Greek, λογικοφανής is a word that describes a person who spits out seemingly logical statements. It often has a negative connotation as it is targeted towards people who tend to fool the public with supposedly beautifully-constructed thoughts that sound convincing in our ears but are not necessarily accurate. The first word that comes to mind is deceptive, but it is not quite what I look for. Any thoughts? How to start applications such as xcompmgr at start up? I've been trying to make xcompmgrto autostart in Lubuntu 11.10 but it just doesn't, I have to start it manually.  I have added the next lines to the autostart script in ~/.config/openbox/  #Run a composite manager xcompmgr &amp;   But whenever I log in it doesn't do anything, I don't really need any special parameters for it I just want it to start at log in.  How can I make it run automatically at login? While I can post external texts into question and answer bodies, it preserves the old font style, spacing and size (although the markdown display won’t be affected by that.)   App Version: 1.6.6.2 Device: iPhone 6 OS Version: Version 11.0.3 (Build 15A432) I am having a problem with Windows 7. When I boot up, the CPU jumps to 100% execution for about 10-15 minutes. During this time, it is really slow, and if I play any music, the songs play slowly and sound distorted.  I tried to scan my computer with different antivirus programs, I used Spybot, HijackThis, and CCleaner, but I didn’t find anything of note.  I watched the process list in Task Manager and saw that there are two processes that are to blame. There are two copies of svchost.exe, one using ~48% CPU, the second (under the Network Service account) using ~37%. For the first 15 minutes, the CPU runs at about 70-80%, then eventually drops to 30-40%.  I don’t know what to do. I could not find a way to stop the services, and if I try to kill them, sometimes the system crashes.