Why should be accept this ${\displaystyle {n \choose k}}=0 $ for $k> n$? I'm confused how  ${\displaystyle {n \choose k}}=0 $  for $k&gt; n$ however the  ratio :${\displaystyle\frac{n!}{(n-k)!k!}}$ is not defined for $k&gt;n$ ?  My question here is: Why should be accept this $${\displaystyle {n \choose k}}=\frac{n!}{(n-k)!k!}=0 ,  for , k&gt; n$$ ?  Note: I don't accept this if it were by convention because it's not defined !!!!  Thank you for any help
$n$ choose $k$ where $n$ is less than $k$ I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.  In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?  Thanks!
How can I create a water material in Cycles? Do WALLS matter if I'm building the 6-inch brick defense from Corruption/Crimson for my city? I'm on a new computer and I haven't downloaded Data Loader in years but now, I'm having all kinds if issues.  So apparently, I have to download Zulu JDK 11 (msi) first, Then download and unzip Data Loaded and install. Which opens a command prompt... After I follow all of the command prompt steps, then I get an error.  Am I supposed to install Java too? And if so, do I download Java or Zulu first?  Apparently a lot of people are having this same problem with no resolve.  Any help would be much appreciated. Labelled spanning trees of $K_n-e$ Let $e$ be an edge of $K_n$- the complete graph on $n$ vertices. Prove that the number of labelled spanning trees of $K_n-e$ is $(n-2)n^{n-3}$.  I think the answer lies in using some modified form of Prüfer's sequence but I can't quite nail it. biblatex custom reference I've been searching for a way to create "Custom References" with biblatex, but haven't found a solution to my problem. Among "normal sources" like Journal Articles, I need other references where I can put a custom sentence in it. Like "[3] This was performed by someone else".  Is there an option to do that?   \documentclass{scrreprt} \usepackage[refsection=chapter,style=chem-angew]{biblatex} \usepackage[english]{babel} \usepackage[T1]{fontenc} \usepackage[ansinew]{inputenc} \begin{filecontents}{biblio.bib}  @article{entry1,      author = "George D. Greenwade",     title = "The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})",     year = "1993",     journal = "TUGBoat",     volume = "14",     number = "3",     pages = "342--351",            url=" www.ctan.org" }  \end{filecontents}  \addbibresource{biblio.bib}  \begin{document} \chapter{Test Chapter One} \cite{entry1} So far so good, now I need another reference that says 'This was performed by someone else'. \printbibliography[heading=subbibliography]  \end{document} How do I change folder timestamps recursively to the newest file? Should I set the title attribute for content DIV's to explain what they contain? I have a number of DIV's that break up a page into logical elements for styles and positioning, etc.  As part of optimizing my site for search engine spiders, should I include a title attribute within these DIV's?  (Yes I have a title as part of the whole page, I'm curious on individual structual elements)  Ex  ... &lt;div id="SearchOptions" title="Search filters and options - Type in a name or pick a business and Click 'Search' to get started" &gt;   .... &lt;/div&gt; ... Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$ Let $f:\mathbb{R}\to\mathbb{R}$ be a function, two times   differentiable with $\left|f(x)\right|\leq1,\forall x\in\mathbb{R}$   and $f^2(0)+\left(f'(0)\right)^2=4$. Prove that there exists at least   one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$.   I have tried the usual Rolle method by $e^x$, but didn't go far... Any help available? I declared random list using TikZ package and can get items randomly. But at same time I need to get all items in sequential order. How can I do it? Is it possible print \mylist items declared in noncomma variat?  \documentclass[a4paper,14pt]{extreport} \usepackage{tikz}   \def\mylist{{one}{two}{three}{four}{five}}   \pgfmathdeclarerandomlist{mynum}{\mylist} %Define the list   \begin{document} \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\ \end{document} What would be a good approach to develop the render system for an oldschool type racing game that uses a pseudo 3D scenery, like for example Outrun or Lotus Esprit Turbo Challenge? There's an endless scrolling road and scenery graphics are placed like billboard items, etc. I think you get the idea.    Are these developed in a similar way like  or is there a different technique behind it? Does the technique has a specific name I could search for on the web? I need to develop something like that in Flash. How to understand linux versioning I need to understand the Linux versioning system and distribution concepts. what are public, stable or final terms for versions? Can I use Gödel numbers to make a proof? Can we make the "expanded user card" privilege universal? Pursuant to discussion on the , several have asked how far they'll need to search to determine if a user has expressed their preferred pronouns. This requested feature could create an easy solution for that concern.  On sites where I've reached the  privilege level, users can hover over my picture to see a snippet of my profile:    This is where I've included my preferred pronouns, and where I see many others including them as well.  By making this feature universal, users could quickly and easily check for preferred pronouns without needing to navigate to another page. Is it possible to serialize and deserialize a class in C++? Why doesn't **sort** sort the same on every machine? Using the same sort command with the same input produces different results on different machines.  How do I fix that? What is a less offensive synonym for "hypocrite"? Is there a word that describe a person who doesn't "practice what they preach"? Basically, is there a synonym for "hypocrite" that carries less pejorative connotations?   For example, let's say a friend of mine says he is a Christian. Yet he engages in behaviors considered sins by the Bible, making him a hypocrite. And I would like to tell him I think he is a hypocrite.   What would be a good word to use here, so my criticism seems constructive rather than offensive?     I tried using a , but the synonyms that came up were even worse (ie: they are more offensive than "hypocrite"). Related (but different) question: , which asks what to call someone who knows they are in the group of people they criticize. How to calculate information included in R's confusion matrix I want know the formulas for the information highlighted in the confusion matrix in the picture below: Which focal-length lens is usually used for portrait photography, and why? What focal length or range of focal lengths are usually used for portraits?   What is used by professionals? Why would one select a certain focal length? What are the advantages and disadvantages of different choices, and are there certain situations where an unusual choice might be better? Is it possible for reviewers to mistakenly reject a quality paper? Here is a simple table.  I want to merge 4 cells. I want to combine the three "this" cells area in this table. I thought I have to use multirow and multicolum at same time. However, I could not find how to use it.  If possible, could you post the modified LaTeX source code?     My LaTeX code:  \documentclass{article} \usepackage{multirow}% http://ctan.org/pkg/multirow \usepackage{hhline}% http://ctan.org/pkg/hhline \begin{document}  \begin{table}[ph] \begin{center} \begin{tabular}{c|c|c|c|c} \hline \multirow{2}{*}{Raaa (k)} &amp; \multicolumn{4}{c}{C ()} \\ \hhline{~----}  &amp; 3.3 &amp; 2.5 &amp; 1 &amp; 0.5 \\ \hline \multirow{2}{*}{Raaa (k)} &amp; \multirow{2}{*}{this} &amp; this &amp; 0.5 &amp; 0.6\\ \hhline{~~~--}            &amp;                       &amp; this &amp; 0.7 &amp; 1.2 \\ \hline \end{tabular} \caption{R, C ripple size} \label{T:peak} \end{center} \end{table} \end{document}