"A Math Club has 12 members. They have to choose euclid contest team   of 3 members and a National Math Competition team with 4 members. Students can be on both teams, but James agrees to   only be on at most one team. In how many ways can both teams be chosen?"   Here's what I thought should be the answer but doesn't make much sense:  Case 1: James is in Euclid team:   $\therefore $  $11\choose2$ for Euclid Team  This would be because James is in team Euclid assumed so we only have to get 2 other players and we have 11 students left.  Also : $11\choose4$ for Math team  so total ways would be  $11\choose2$ . $11\choose4$ (Product Rule)  Case 2: James is in Math team:   $\therefore $  $11\choose3$ for Math team  This would be because james is in team Math assumed so we only have to get 3 other players and we have 11 students left.  Also : $11\choose3$ for Euclid team , as the 11 students can form a team of 3 for Euclid competition here.  so total ways would be  $11\choose3$ . $11\choose3$ (Product Rule)  Case 3: James in no team  $\therefore $  $11\choose3$ . $11\choose4$ (Product Rule)  as James is in no team so we are left with 11 people for both teams  Now the question is did I do it correctly? Because I have never seen cases being used in a permutation questions?  Therfore, The total number of ways would be to add all the cases meaning  Case 1 + Case 2 + Case 3 :  ($11\choose2$ . $11\choose4$) + ($11\choose3$ . $11\choose3$) +($11\choose3$ . $11\choose4$)  Is that correct?
A club has $12$ members. They need to choose a programming team of $3$ members and a hacking team of $4$ members. Members can be on both teams, but Jack only wants to be on at most one team. How many ways can both teams be chosen?  I did two different ways. I solved if Jack chooses programming, $C(12,3)=220$ $C(11,4)=330$. In total there are $550$ different ways both teams can be picked.  But, if Jack chooses hacking, I solved $C(11,3)=165$ and $C(12,4)=495$. In total there are $650$ ways both teams can be chosen.  I was wondering if this is the correct method.
How does one attack a two-time pad (i.e. one time pad with key reuse)? My question might appear the same as the question , but actually I did read all the answers and none of them helped me with the details I need.  I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt.  I know that XOR-ing two ciphers gives me the XOR of their original messages.  My question is what is the correct thing to do after that?  I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$.  Then get $S_1 = C_1 \oplus C_2  \oplus $' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '.  After that I compared all corresponding characters in $S_1$ and $S_2$,  and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$.  I tried this on paper before coding and it worked, but I might be missing something.  Is this the right approach? Why does it work? I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. Is posting a very long quote OK for an answer? I just found  to a question on SO. While it provides a source for the (very long) quote, I do not like the fact that the answerer just posted the quote, without adding anything. Maybe add highlights/emphasis or remove uninteresting passages. Or skip the quote altogether and write a short summary. I feel, in the current form, this should have been a comment that just links to the original article.  Am I too pedantic? Should I flag this as not an answer? Should I downvote it? What do you think about such answers? Increasing the Size of Beamer Footer I work in Beamer extensively (specifically the Madrid theme). Sometimes, my "title" is quite long. Beamer puts this title at the bottom, but when it's too long it cuts it off. I'd really like to know how to increase the size of the bottom middle rectangular footer.    Here's the code (sorry for all of the packages and commands):  \documentclass[t]{beamer} %Add "mathserif" to script math lettering. "t" places all text to start at the top of each slide \usetheme{Madrid}  %Packages  \usepackage{amsmath} %Math mode \usepackage{amssymb} %Math mode \usepackage{commath} \usepackage{graphicx} %Including pictures \usepackage{color} %Define colours \usepackage{setspace} %Enables double-spacing/variable spacing \usepackage{pgfplots} %Draw graphs \usepackage{upgreek} %Insert theta symbol as unitalicized  \usepackage[version=3]{mhchem} %Chemistry equation mode \ce{} \usepackage[export]{adjustbox} %Place black borders around images \usepackage{scrextend} %Used to change font size \usepackage{caption} %Used to utilize * in caption function \usepackage{subfig} %Used to place figures side-by-side with sub-captions \usepackage{siunitx} %SI Units (+ scientific notation) \usepackage{cancel} \usepackage{fancybox} %\usepackage{enumitem} %Making lists %\usepackage[english]{babel} %Lists with letters  %Erases Beamer Navigation Symbols on the bottom left \setbeamertemplate{navigation symbols}{}  %\setbeamertemplate{footline}{} \captionsetup[subfigure]{labelformat=empty}  %Defined Colours \definecolor{orangered}{RGB}{255,69,0} \definecolor{darkgreen}{RGB}{0,100,0}  \definecolor{firebrick}{RGB}{178, 34, 34} \definecolor{navy}{RGB}{0,0,128} \definecolor{lightsteelblue}{RGB}{176,196,222} \definecolor{steelblue}{RGB}{70,130,180} \definecolor{darkslateblue}{RGB}{72, 61, 139} \definecolor{forestgreen}{RGB}{34, 139, 34} \definecolor{Gold}{RGB}{218,165,32}  %Defined Colour Theme and List Sizing \setbeamercolor{structure}{fg = Gold} \setbeamertemplate{navigation symbols}{}  \setbeamerfont*{itemize/enumerate body}{size=\normalsize} \setbeamerfont*{itemize/enumerate subbody}{parent=itemize/enumerate body} \setbeamerfont*{itemize/enumerate subsubbody}{parent=itemize/enumerate body}  %Opening Title Page \title{Chapter 7 - Absolute Value and Reciprocal Functions} \subtitle{Section 7.3 - Absolute Value Equations} \date{}  \begin{document}     %Slide 1     \maketitle \end{document} How can I disown a running process and associate it to a new screen shell? I have a running program on a SSH shell. I want to pause it and be able to unpause its execution when I come back.  One way I thought of doing that was to transfer its ownership to a screen shell, thus keeping it running in there.  Is there a different way to proceed? How do I connect to a WPA wifi network using the command line? So basically, I'd like to use iwconfig to connect to my wifi network when I'm not inside X. But I just don't see a way to do it in the man page. Any idea? Browse local copies of websites when possible (Firefox with slow internet connection) I am on a slow internet connection. I would like to use Firefox to load local copies of websites when it exists and is relatively recent (say one week).   An example usage would be browsing a lot of Wikipedia webpages, coming back to the same ones on a regular basis.  I would like to avoid having to manually specify to save single pages on each website I visit (which is what I guess I would need to do with a plugin like ). And I would also like to avoid having to switch between "work offline" or "work online".   Any suggestions? We are experimenting with running an OpenVPN server for our business. One question I can't seem to find the answer to is this:  When we generate keys for one of our users for them to use at home, can their use the same keys on their home laptop as well as their home desktop? Or do we need to generate separate keys for each user's client machine? Is it possible to search for down voted questions? I want to search for new downvoted questions to improve or flag them.   I tried in the searchbar: votes:-1 is:question   but votes:XX only works with positive numbers. I can't query for downvoted questions. Is there a work-around to this? If there's not, should this feature be added? Can we clarify to the OP that their question is not yet closed and the duplicates are only suggestions? I keep seeing custom flags in the queue from question askers who are incredibly irritated because there is a big, prominent box at the top of their question. It's just a list of other questions which may have the answer they desire, and is only even displayed to them because the question is not closed yet.  This question may already have an answer here: [list]  But to an ordinary user, it is not at all clear that the box is only shown to them, or that it will go away on its own if their question never actually gets closed. Could the text for this notice be altered to make it more clear that:  They are just suggestions for the person to view. Potentially if they agree with one, they should vote to close their own question as a duplicate of it.  Their question is not closed yet. If they disagree with the duplicates suggested, they should clarify their question and explain why it's not a duplicate.   I was thinking of an altered wording such as:  You might be able to find an answer to your question here: [list] If those answers do not fully address your question, please [edit your question] to clarify your exact problem.  I went with a more direct wording for the bolded title. Since the notice is only displayed to the asker, it makes more sense to talk directly to them in the message rather than making it feel like it's supposed to be addressing the entire audience of the site. Emphasize that only that person is seeing it by talking to them. It would also be helpful to append a &quot;More info&quot; link at the end of the message that links them to some further documentation about duplicates, and explains how the notice works - particularly that it will go away eventually or change to a proper duplicate notice if the question does get closed. How does ZFC describe addition? Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked  elsewhere online for it, but didn't find it.   Intuitively, the cardinal of the union of two sets seemed appealing. But that approach doesn't work, given that the union of 2 and 2 is 2. I wondered whether ordered sets might be essential to the answer.   So, how does ZFC describe addition? How do I get current date/time on the Windows command line in a suitable format for usage in a file/folder name? "Back" and "Install Now" buttons grayed out in 14.04 installer The "Back" and "Install Now" buttons are grayed out in the 14.04 installer. How do I restore them? Instead of the constrained 45 degrees that happens when you press C, is there a specific angle I can get for example 35 degrees instead?   It's currently cut at 45 degrees in the attached image and I need it to be 35. I have a video that was rotated 180° when recorded. Is it possible to correct this with FFmpeg? Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$. Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$.  I showed that this series point wise.  Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |x|^n$$ converges, By abosultely convergese test $\sum (-1)f_n(x)$ converges.  I want to show that this series uniformly converges on $[0,1]$. Is $\int \frac{1}{2x} \, dx $ equal to $\frac{\ln|2x|}{2}+ C$ or $\frac{\ln|x|}{2}+ C$ Reasoning for $\frac{\ln|x|}{2}+ C$ :  $$\int  \frac{1}{(2x)}  \, dx = \frac{\ln|2x|}{2}+ C$$   Check: By taking derivative we get $\frac{1}{x}$  Reasoning for $\frac{\ln|x|}{2}+ C$ :  $$\int  \frac{1}{2x}  \, dx = \frac{1}{2}\int  \frac{1}{x}  \, dx =\frac{\ln|x|}{2}+ C$$  Check: By taking derivative we get $\frac{1}{x}$  Question: Is  $\int  \frac{1}{2x}  \, dx $ equal to $\frac{\ln|2x|}{2}+ C$ or $\frac{\ln|x|}{2}+ C$? Why does chmod 777 -R / leave the system unusable? Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Add a non-breaking space to "reopen (1)" and its ilk