Problem  $A = \{1, 2, 3, 4\}$ $B = \{1, 2, 3, 4, 5\}$ Find the number of mappings from A to B such that $f(i)\le f(j)$ for all $i&lt;j$.  My approach: The required set of functions= $\{\text{Set of total possible of functions}\}\setminus\{\text{Functions such that $f(i)&gt;f(j)$ for $i&lt;j$}\}$   (i) The number of possible functions would be $5 \times 5 \times 5 \times 5 = 625$.(There are $5$ mappings possible for each one of them.)  (ii)To find the number of functions such that $f(i)&gt;f(j)$ for $i &lt; j$, I can choose any four elements from $B$ in $C(5,4)$ ways. This will have only one decreasing order and only one arrangement is possible out of every set of elements I will choose. I will therefore have $5$ such functions.  So the required number of functions would be $625-5 = 620$ which largely differs from the answer stated by my textbook, i.e. $70$. Please help me out.
I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance.
Text not very readable on different image backgrounds I've come here seeking for a bit of design help or advice on how I could possibly improve the following:  Like the title says, I'm having a website banner with multiple text displayed on different images as backgrounds. Now, since most of the images are dark, I've set the font color as white. But even then, some of the text is not very readable on a couple of images like:    Since I'm working with CSS, I've tried different options such as:  Adding Text Shadows:    CSS:  text-shadow: 3px 3px 0px #000;  Also tried, creating a semi-transparent box around the text:      I feel the box one looks out of place.  I've created a  for anyone who knows CSS. If not, please feel free to advice me based on your imagination alone.  PS: You could also advice on using a different font if that would help it stand out better. How to use siunitx with non numerical values? How to add a guest account without a password? A Polygon is inscribed in a circle $\Gamma$ A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove that  $MA\cdot MC=MB^2-AB^2$  I inscribed the polygon P in the unit circle and let B=1. Past this, I'm not really sure how to proceed. It might be helpful to have M at 1 as well but given that the polygon would have the n-th roots of unity in it B seemed like a good choice. What would be a good way to proceed from here? Recently I have learned how to draw simple stuff with TikZ. I really like it. Now I wonder if it is possible to draw with TikZ on a image. Instead of pasting a image in Illustrator and adding some vectors to point something in the picture I would like to do it using TikZ. Is it possible?   For example, on a picture of a computer (PNG,etc) I would like to draw some arrows and labels saying "Keyboard", "Monitor", etc. When I log into my web server via SSH I see the information:  88 packages can be updated. 80 updates are security updates   I tried apt-get update then apt-get upgrade but each time I log in I still see the message about updates. How do I install them? Postfix SMTP banner show multiple hostnames We have a Postfix SMTP server that's bound to two IP addresses and serves mail for two different domains. We can't change the domains to both use the same, single IP. Both IP addresses have reverse DNS   1.1.1.1 reverses to mail.domain1.com 2.2.2.2 reverses to mail.domain2.com   In our Postfix config I have  myhostname = mail.domain1.com myhostname = mail.domain2.com  inet_interfaces = 1.1.1.1, 2.2.2.2  smtpd_banner = $myhostname Mail Server   (And some other stuff, which I think is irrelevant)  Using , running the smtp test command on mail.domain2.com returns everything as OK, however running it on mail.domain1.com returns an error because Postfix is identifying itself as 'mail.domain2.com' in the SMTP banner, how do I get it to return the correct banner based on the external IP address used? Installing TTF fonts in LaTeX One of the annoying aspects of LaTeX is the limited number of fonts that come by default, and the pain involved in making new fonts 'LaTeX' ready. I have a collection of truetype fonts that I'd like to prepare for use, and I definitely want to make sure I have vector versions of these fonts (i.e not type 3/bitmapped versions). Is there a relatively painless way to do this ? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. Why does Beamer not change the itemize symbol when using \setbeamertemplate? This code:  \documentclass[10pt]{beamer} \setbeamertemplate{itemize item}[circle] \setbeamertemplate{itemize subitem}[triangle] \setbeamertemplate{itemize subsubitem}[square]   \begin{document} \begin{frame} \begin{itemize} \item first level     \begin{itemize}     \item second level         \begin{itemize}         \item third level         \end{itemize}     \end{itemize} \end{itemize} \end{frame} \end{document}    produces this output:    I expected the symbol of the first level item to be a circle, not a square. Is there an error in my code? The log file was too long to include in the question, so I . Why isn't The Flash unstoppable? I've recently gotten into The Flash TV show, and so I've been reading about his powers. What I don't understand is why his typical rogues pose a threat - it seems like there should be very little limit to his powers.  He can go back in time, so if he loses a battle or the bad guys do something, he can go back and preempt it. In the new 52, he can all but predict the immediate future, so he can see bad things that are going to happen and stop them. He can phase matter through matter, or let matter pass through him. This means he is all but immune to weapons fire.  I haven't read enough of the old comics so I don't the know challenges he's had. For a guy who can't be attacked, can time-travel and can race around the globe several times in a minute, I can't imagine there is much that could pose a threat. I am considering to create a wiki/blog for most of my study notes that I have made or taken digitally.  Also, for the handwritten and rough ones I am thinking about storing them in my GitHub repository.  Is this a good idea? How do professionals in academia and PhD students organise their notes?  Also, should I keep my notes publicly available?  Edit: The main purpose of keeping my notes public is to have them readily available and searchable of myself and my immediate peers in college. If that benefits someone else, I will be more than glad to have helped.   Update regarding having a blog/wiki: I was thinking about having stuff regarding my take on a certain topic or similar rather than just straightaway putting mugged up lecture notes. (as I mentioned in a comment)  So basically, as I go along a certain topic whatever interesting I find, I post  it. Also I will put in links to papers, books and other interesting literature on the topic that might be helpful while studying the topic. This seems easier to deal with and also serves as a way to track my current interests and/or workflow. ADC0808 IC working correctly occasionally I am using  for a project on converting analog signal to digital signal. It has an inbuilt multiplexer using which we have to provide an analog signal to one of the pin. However what happens is ;  Sometimes it latches to a particular value say 1000000 and remains at there no matter how many times Start of Conversion is pressed which should set the inner counter to 0.  Sometimes it gives slightly wrong output off by 100 or 1000 and when I press Start of conversion, it gives correct output, so its not reliable every time.  And most seriously , Suppose the control for multiplexer according to Datasheet are such that input pin 1 should be active, instead, it works only when the input is given to pin 5 for one particular IC and pin 3 for other. I have been trying to fit it in logic since 3 days however each day some new error happens. I have replaced it twice from my college lab and now the technician doesn't want to replace it anymore . What is amiss here ? Is there any bidirectional 5v-3.3v level shifter? Is there any IC that interfaces any 3.3V Input/Output to 5V Output/Input? I need it primarily for an Arduino Due but if there any bidirectional IC that works like that it would be great.  Some people advised me to use the SN74AHC125 and CD4050 ICs, but I don't understand how they work or how to interface with them. How to contact Canonical? So I am doing a College research project for my Computer Information Systems department and it is about alternative Linux server distribution for small businesses. From my findings I would really like for this research to actually take off into a Distribution project (Using Ubuntu as a base OS).   However I would like to rename the distribution as "Bizbuntu" or something related to that ("Biz" being shorthand for business) however I don't know if I can name the distribution without consent from Canonical.   So my question is do I have to contact Canonical about this, if so, does anyone know a mailing list or email that I can contact them about this? Is there a function to round a float in C or do I need to write my own?       float conver = 45.592346543;   I would like to round the actual value to one decimal place, conver = 45.6. I have been playing on hardmode for a while and have beaten the three  a couple of times. But I have yet to find a . It's getting really frustrating.  Does anyone have any methods to finding a Plantera Bulb? Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$ I recently came across this problem     Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times  }} = 3$   After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists.  So my question is     Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges.   Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help .  If the solution from that book is required I can post it but it is not complete as I mentioned.   Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist. Creating fillable PDFs I'd like to create a LaTeX document that when rendered into PDF, has forms that can be filled out using Adobe Reader or other such programs. Then I'd like to be able to extract the data. I deliberately would like to avoid using Acrobat for all the usual reasons (non-free, need different versions for different platforms etc).   Can this be done ? Isn't every subset of a compact space compact? Let $\mathfrak{C}$ be an open cover of a topological compact space $X$, and let $\mathfrak{B}\subseteq \mathfrak{C}$ be its finite subcover. Then every subset of $X$ also has the same cover and subcover! Shouldn't every such subset also be compact then?  Motivation: I read somewhere that every closed subset of a compact space is compact. I wonder why it shouldn't be the case for every subset of $X$, rather than just the closed ones.