I've been trying to solve this problem, but i don't know what technique i should be applying and my answers are coming out to be wrong. I've tried putting them in different groups but non of them give me the answer.  Question: There are 10 people standing in a straight line. Find the number of ways to choose 3 of them such that the no 2 of them consecutive.  Please help!
During self-study, I ran across the question of how many ways six numbers can be chosen from the numbers 1 - 49 without replacement, stipulating that no two of the numbers be consecutive.  I can obtain a simple lower bound by saying that, in the worst-case scenario, when you choose a particular number, there are now three numbers that cannot be chosen next.  For example, if I first pick 17, then I can't choose 16, 17, or 18 for the remaining choices.  This gives me the lower bound  $$\frac{49*46*43*40*37*34}{6!} = 6,773,770 \frac{8}{9}$$  This is about 48% of ${}_{49}C_6 =  13,983,816$.  The real answer must be bigger (and an integer).  I haven't found a way to calculate it, though.  The original problem asked to show that the probability of having non-consecutive integers when you choose six is greater than 50%, so if the problem is complicated to count exactly, better approximations that show the answer is above 50% would also be appreciated.   Of course, I can use a computer to do the counting, but I'm interested in learning what methods I'm missing.
What did Hisoka do in the Hunter Exam? In the first Hunter Exam of the series (Hisoka's second),  was introduced into the series removing another applicant's arms from his body for bumping into him and not apologizing.    What did he actually do? We know Hisoka's abilities are Bungee Gum and Texture Surprise, but they don't explain how the arms are converting into flower petals as seen above. Canonical URL for a home page and trailing slashes How do I reboot a Ubuntu Server from the command line? sudo reboot now never seems to work for me. Because its a server, I don't have a window manager installed, so I can't do it from a GUI (which always seems to work).    I've had the problem for years on different versions of the operating system. It happens in virtual machines and on real metal. The same problem exists with sudo shutdown now, too.  I've lived with it in the past. The difference now is I don't want to just hit the power button because I might lose data. Why cannot a set be its own element? Read from location on console C# How to reset or recover admin account password for MySQL? I have a MySQL database that I "inherited" and I was not given the admin credentials.  I do however have access to the box that it runs on.  Is there a way to either recover the admin credentials or create new ones? 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} Post hoc test after ANOVA with repeated measures using R Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$ Let $U\subset\mathbb R^3$ be an open set, and $f:U\to \mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}(\{0\})$ is non-empty, and that at each $p\in S,$ the gradient $\overrightarrow \nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $p\mapsto \overrightarrow \eta(p)=\frac{1}{||\overrightarrow \nabla f(p)||}\overrightarrow \nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $x\in U,$ write $H(f)_{(x)}$ for the $3\times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=\frac{\partial^2f}{\partial x_i\partial x_j}(x).$$   Show that, at each $p\in S$, the second fundamental form $II_p: T_p(s)\times T_p(s)\to \mathbb R$ is the symmetric bilinear map    $$II_p(\overrightarrow v,\overrightarrow w)=\frac{-1}{||\overrightarrow \nabla f(p)||}\overrightarrow v\cdot H(f)_{(p)}\overrightarrow w,$$for all $\overrightarrow v ,\overrightarrow w \in T_p(s)$.       (Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span\{ {\overrightarrow \eta(p)}\})^{\bot}$ of $\mathbb R^3$.      Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.      Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-&lt;d  N_p(v),v&gt;$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)\to T_p(S)$ is the differential of the Gauss map.   Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.   I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...  Looking forward to an understandable explaination. Thanks in advance.     Edit 2:Furthermore, show that, at each point $p\in S$, the expression   $$\phi_p(z)=det\pmatrix{-H(f)_{(p)}-zI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0}$$   (the underlying matrix here is $4\times 4$) defines a second-degree polynomial whose roots $\lambda_1$ and $\lambda_2$ are $||\overrightarrow \nabla f(p)||k_1$ and $||\overrightarrow \nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.      Also, if a non-zero vector $\pmatrix {\overrightarrow v \\c}$ lies in the kernel of the $4\times 4$ matrix $$\pmatrix{-H(f)_{(p)}-\lambda_jI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0},$$   then $\vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$. I'm looking for a way to have two separate objects be connected, so that when I move the first object the second one will move also. The term parent is used, but how do I parent a object? PostgreSQL: running count of rows for a query 'by minute' Save an entire webpage, with all images and css, into just one folder and one file, with wget? Is it correct to hyphenate with compound premodifiers? If so, where is the hyphen placed? For example, "file system" and "related". Is it "file system-related"? It will appear as if it is a compound of "file" and "system-related", won't it? I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook. Why does chmod 777 -R / leave the system unusable? I'm only granting permission to all to do anything but why does the system crash by giving permissions only? I'm only modifying the permission not changing the files. Windows Update Slow on Vista Update I was able to get Readiness Tool to run but not the update Now the reediness tool will not even run Security Essentials will run    I have legal Windows 7 to install Do you think that is safe bet to try?    Had some performance issue so I ran MalwareBytes and Security Essentials.     It found some malware that was successfully removed.    I was able to Update Security Essentials and I ran it again for safety and it found nothing additions.     The last successful Windows Update was 5 days ago     Checking for Updates has been running 6 hours CPU 100%  network 0%     Windows6.0-KB947821-v35-x86.mnu fails to install It is supposed to be a tool to address update problems   I fear the registry got hacked to prevent updates     What can I do?  I know Vista.  I got it free. Which Pokémon can speak with humans? Inspired by Detective Pikachu, I was wondering which Pokémon can "speak" with humans.  I would like to include  Pokémon like Meowth in the anime that speak English/Japanese with their mouths, Pokémon like Mewtwo in the anime/movies that speak telepathically, or other Pokémon we've seen like Lugia/Entei that seem to speak without moving their mouths but still out loud for all to hear (presumably).  The format doesn't matter (manga, anime, video games). And while this is inspired by Detective Pikachu, it's not about it, as the movie is not out yet      And it seems like the human character has the special ability, not the Pikachu My keyboard doesn't have an Fn key so I can't get the coordinates to show up. I don't know how to get into the multiplayer settings and Shift + F3 doesn't work either.  How can I fix this? I often use  to keep my sentences gender neutral. Many of my friends (none of them are native speakers), however, consider this to be grammatically incorrect, and suggest me not to use it in formal English exams.  My questions are:   Is singular they widely accepted in English speaking countries such as America and Britain? Is it formal enough to be used in formal situations, such as a public speech? Is it casual enough to be used in informal situations, such as gossiping among friends?