No of ways of making Garland using different kinds of flowers Question: There are 6n flowers of one type and 3 flowers of another type, total no of garlands possible?  My approach-: I have considered 3 cases Case 1. Garland contains only 6n  flowers No of possible ways $= \frac{(6n-1)!}{2}$  Case 2. Garland contains only type containing 3 flowers No of permutations$= \frac{2!}{2}$  Case 3. Garland has both types of flowers No of permutations $= \frac{(6n+2)!}{2*6n!*3!}$  Total permutations$= (6n+2)!/(2*6n!*3!) + 1 + (6n-1)!/2$  Is this approach correct or have I missed/considered some cases? What approach should be used for such questions?
How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers? This is my first question here.   I'm given $3$ white flowers and $6m$ red flowers, for some $m \in \mathbb{N}$. I want to make a circular garland using all of the flowers. Two garlands are considered the same if they can be rotated to be the same. My question is: how many different garlands can I make?  I tried an approach in which I divided the question into three parts:   No. of ways in which the white flowers are all together No. of ways in which two white flowers are together No. of ways in which no two white flowers are together   Can you please help?  The answer is $3m^2+3m+1$ in the book that I saw  Thanks
Determining the structure of the quotient ring $\mathbb{Z}[x]/(x^2+3,p)$ What is the difference between correlation, causation and prediction? Suppose we have a set of events $\Omega$, containing events $A$ and $B$. My econometrics professor tried to distinguish the following three terms today.    Causation --- $A$ causes $B$ if the occurrence of $A$ always leads to another specific outcome $B$. For example, clapping my hands causes a sound to be emitted.  Prediction --- $A$ predicts $B$ if on average, $B$ is the expected outcome from $A$ occurring. In other words, whereas causality is deterministic, prediction is probabilistic. For example, studying for a test would predict doing well on the exam. But there's no guarantee.   Correlation --- $A$ and $B$ are correlated if when one occurs the other does too. But there may be some hidden variable causing both.    Are these distinctions meaningful? Or did he make these terms up?   My Questions   What is the difference between causation and correlation?  Are "causes" always deterministic? Here is the documents needed.     Here are the documents needed.       Which is grammatically correct and why? My guess would be the second one because of the plural form. Should I submit bank statements when applying for a UK Visa? What do they say about me? The UK Visas and Immigration Directorate publishes  that provides helpful tips that can result in successful visa applications. Section 2 of this guidance opens is entitled &quot;other documents you may want to provide&quot; (emphasis mine).  Section 2: other documents you may want to provide – all visitors This section provides guidance on the types of documents that you may want to provide to help us consider your application against the Immigration Rules. Previous travel documents/passports, which show previous travel. Financial documents showing that you have sufficient funds available.  These must clearly show that you have access to the funds, such as:  bank statements building society book proof of earnings such as a letter from employer confirming employment details (start date of employment, salary, role, company contact details) where a third party (who is either in the UK or will be legally in the UK at the time of your visit) is providing financial support to you e.g. a business, a friend or a relative, documents to show that they have sufficient resources to support you in addition to themselves and any dependant family should be provided   The first item is &quot;bank statements&quot; and this ties in with the , which says...  evidence that you can support yourself during your trip, eg bank statements or payslips from the last 6 months  (Note 'eg' is a Latin abbreviation meaning &quot;for example&quot;; this appears to be optional rather than a fixed requirement.) Question: How should someone interpret this such that the chances of success are maximized? How many statements should I submit? What are they looking for and what do my bank statements tell about me? Find the infinite sum $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $ I am facing a problem, where I have to find the partial sum of a sequence/sum and with that, the infinite sum of the sequence.  $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $   The problem here is that I don't know how to proceed. I would be thankful if someone would steer me in the right direction. Why is the node editor showing different kinds of nodes? Kinematic Viscosity How would you define ? What does it physically represent? Around the Internet I've found it defined as just a ratio, and that's it.  I saw in an answer that I can think of it as "diffusivity of momentum". What does this mean? Grub does not show a Windows 8 option after dual boot There has been quite a few times that I wished I could send a message to another user on SO - not ask a question for everyone to see, but just a short message informing them of something or requesting them to do something.  Are there any plans to allow this to happen in the future?  Related: Travelling from India to canada via Dubai and London. Do I need a transit visa I am embedding a PDF file inside my LaTeX-generated PDF file using embedfile:  \usepackage{embedfile} \embedfile{myfile.pdf}   I would like to also insert a link that the user can click on to open the attachment.  Is there any way to do that? "In and of itself" is a phrase I find myself using all the time. But in and of itself, the phrase "in and of itself" has no meaning. That is, the individual words don't seem to contribute to the whole meaning of the phrase. Introduction to statistics for mathematicians What is a good introduction to statistics for a mathematician who is already well-versed in probability?  I have two distinct motivations for asking, which may well lead to different suggestions:   I'd like to better understand the statistics motivation behind many problems considered by probabilists. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.   I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source. Restrict node list in entity reference field based on node field My "Article" nodes have a "domain" field that links to a taxonomy term. My "Article" nodes also have a repeatable "entity reference" field "more like this" that links to other "article" nodes.  How can i restrict the list in the "more like this" field to show only the nodes that have the same "taxonomy term" that the node i am currently editing ?  Thanks a lot ! How can I drive 12v LEDs from my Arduino Nano? Background I am building a scale RC helicopter and ordered a bunch of 3mm LEDs to give it some nice navigation and strobe lights which I plan to drive with an Arduino Nano. Problem is that I goofed and didn't notice that the LEDs are 12v until they arrived.  I know that the Arduino Nano can accept 12V on pin 30, but I think the voltage output on any of the &quot;D&quot; or &quot;A&quot; pins is always 5v (correct??). Question If the voltage output of the Arduino pins is 5v, then how can I drive the 12v LEDs using the Arduino, or is it impossible? Clipboard.GetText returns null (empty string) My clipboard is populated with text, but when I run  string clipboardData = Clipboard.GetText(System.Windows.Forms.TextDataFormat.Text);   I get back an empty string. I've toyed with various forms of the call including:  string clipboardData = Clipboard.GetText(); string clipboardData = Clipboard.GetText(System.Windows.Forms.TextDataFormat.UnicodeText);   But with the same result.  Am I missing something obvious? A question here on meta for instance may stick around for quite a while. A feature-request  maybe even for months. Then the status (tags) gets changed to  or  or whatever and the OP does not get notified.  On a normal post-edit we get a notification but retagging does not count for that. Please add retagging to the edit notifications. Laggy viewport in 2.9 when model has a lot vertices What is a "static" function in C? The question was about plain  functions, not  static methods, as clarified in comments.  I understand what a static variable is, but what is a static function?  And why is it that if I declare a function, let's say void print_matrix, in let's say a.c (WITHOUT a.h) and include "a.c" -  I get "print_matrix@@....) already defined in a.obj", BUT if I declare it as static void print_matrix then it compiles?  UPDATE Just to clear things up - I know that including .c is bad, as many of you pointed out. I just do it to temporarily clear space in main.c until I have a better idea of how to group all those functions into proper .h and .c files. Just a temporary, quick solution. Space of bounded continuous functions is complete I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete.  The lecturer then proved two things, (i) that $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in \mathbb{R}$.  I don't understand why it's not enough that $f$ is bounded. I think the limit of a sequence of continuous functions is continuous and then if $f$ is bounded, it's in $C_b(X)$. So what is this $\lim f_n \in \mathbb{R}$ about? Many thanks for your help.