Can someone explain to me the proof of $${n+1\choose k} = {n\choose k} + {n\choose k-1}$$?
I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas?
What is the difference between the memory usage report in System Monitor and the one by free? Are there any tools, methods, incantations to recover recently deleted files on Ubuntu?  If it makes any difference, I want to recover a  2.x database file. But would be better to have a method/tool that works on any kind of file. Can different gauge wiring be used in the same circuit? What is "anycast" and how is it helpful? I'd never heard of anycast until a few seconds ago when I read "".  The wikipedia "" article on it is quite formal and doesn't really evoke a mental picture of how it would be used.  Can someone explain in a few informal sentences what "anycast" is, how you configure it (just in a general sense), and what its benefits are (what does it make easier)? Shouldn't rep gains from old questions & answers eventually stop (or at-least reduce)? QR code generation library in javascript A sci-fi book or TV show about a tribesman discovering he or she actually lives in a huge spaceship Could a website force the browser to go into fullscreen mode? How does order-preserving encryption work? Order-preserving encryption (OPE) is, apparently, a method of encrypting data so that it's possible to make efficient inequality comparisons on the encrypted items without decrypting them.  I've been coming across this term in various places (including ) lately, but I have no idea how such encryption schemes are supposed to work.  Any obvious methods I can think of to allow such comparisons on encrypted data would result in catastrophic security failure.  Certainly, I assume that OPE must involve some kind of security tradeoff compared to traditional encryption, but given that it seems to be actively  and , it seems that it must be possible to implement in a way that retains at least some level of useful security.  I just don't see how.  Given that  didn't turn up any convenient Wikipedia articles or other popular summaries of OPE, I figured I'd try asking for one here before diving into the academic literature.  (At worst, I may try to answer my own question later, if nobody else beats me to it.)  So, to summarize, my question is: How does order-preserving encryption work, and what security properties does it provide? Recommended visualization libraries for standalone applications How to update kernel to the latest mainline version without any Distro-upgrade? When is the union of a family of subspaces of a vector space also a subspace? It is not difficult to prove that the union of a chain (or, more generally, a directed family) of subspaces of a vector space $V$ is a subspace of $V$.   Given a family $\mathcal{F}$ of subspaces of a vector space $V$ such that the union of $\mathcal{F}$ is a subspace of $V$, is it true that $\mathcal{F}$ is a directed family?  If not, is there a "nice" characterization of families of subspaces whose union is a subspace? This error occurs whenever I try to work with sudo. I changed the permissions of /var/ folder to 777 and I think this causes the error. Please help me get back a functional sudo and /var. How to specify password in ssh command From the terminal I type: ssh user@ip and then it prompts for a password.  Is there a way to specify the password in the ssh command itself? Why do games run slow on CPU Maybe my question sounds naive because I'm not a game developer but I dont see objective reasons for this. An average cpu is in times more powerful than an average gpu like a few graphics cards in one. The most repeated reasons in internet:  the GPU isn't as busy as the CPU  If the gpu renders the screen how it can not be busy The modern cpus are powerful enough. They are not busy most of time. For example my cpu's usage is 3%.  many GPU's are designed to process graphics commands in parallel, CPU single threads them  What bothers the games to run a few parallel processes even on one cpu core? In my opinion its not correct to tell that the gpu is better if games never use 100% of the cpu's capacity. To be specific they dont use even 50%.  To clarify my question based on responses in the comments:  Did you mean to ask &quot;Why do games render more slowly on the CPU?&quot;  Yes I mean why the cpu cant do the whole work for games when it is underloaded and especially if it does the largest part of work in any case.  You can not compare GPUs and CPUs directly, because they do different things.  What bothers them to do the same things?  The CPU isn't as well suited to performing rendering calculations  I dont understand what &quot;suited&quot; means. And why the cpu is not suitable for primitive tasks aka paint roller.  adding additional parallel processes doesn't improve performance  Then why do you produce multicore graphics cards and multicore central processing units? Just give various parts of one task to a few processes if one process is slow and 100% of the capacity is not used. What bothers to emulate the gpu? Its also unclear for me why when you play without a separate graphics module the games see only one cpu core when you play with the gpu they see and use all the cpu cores. May be some magic. Letter class alternatives with support for figures, tables, and labels How to add a directory to the PATH? Let $K\subseteq \mathbb{R}^{n} \times \{ 0 \}\subseteq \mathbb{R}^{n+1} $ and $v=(0,0,\cdots,1) \in \mathbb{R}^{n+1}$.      For every $x \in K$,  let $$Lx=\left \{ tv+(1-t)x \; | \ t \in [0,1] \right \}$$      Define    $$ \text{Cone}(K)=\bigcup_{x \in K}Lx. $$   namely the union of all segments from $v$ to every point of $K$.      According to all definitions that I see on books, Cone is defined:       However the exercise I'm working on  asks to find under what circumstances  $$\text{Cone}(K)=\bigcup_{x \in K}Lx$$ is isomorphic to $(K \times [0,1]) /(K \times \{ 1 \})$ (quotient space).   For example if $$K=\{(1/n,0):n\in\mathbb{N}\} \cup (0,0)$$ I can see both of sets form the same graph but I'm still unable to derive a homeomorphism between these two objets. I already have a 6 month valid UK business visa received prior to April 24 2015. I was supposed to travel to the UK in April, but my trip was cancelled on medical grounds.  I am now planning to visit the UK in July for tourism. Would it be fine for me to do so? Will I be allowed to enter the UK? Should I carry the original invite letter and medical certificate to prove why I didn't go the first time? Cartesian products and families I have some trouble trying to understand Halmos' explanations about generalized Cartesian products. I've read these entries already:    This also has the page I'm stuck on.      Is his generalization the same as a Cartesian product of several sets? As here:   What puzzles me is $\{a,b\}$. Where does it come from, what does it do? I know that it is supposed to be arbitrary. How does it identify distinct elements of $X \times Y$? Does some function map each $a \in A$ to an element in $X$? Or is it always the same $a$?  What does he mean by      the set $Z$ of all families $z$, indexed by $\{a,b\}$, such that $z_a \in X$  and $z_b \in Y$   ? What does does an element of $Z$, namely one family $z$, consist of? By family, does he mean an indexing function or the indexed elements? This comes to a very important question for me: Does family mean a function or the set of all indexed elements?   I will try to explain how I understand it thus far: $Z$ contains elements of the form $\{(a,x),(b,y)\}$. The choice of $x$ depends on $a$, so an $x$ in a $z \in Z$ equals the $z_a$ of $X$. Now, we want to map these pairs to $X \times Y$, so we have a function $f$ that takes $z \in Z$ and connects it to the corresponding ordered pair $(z_a, z_b)$, where $z_a \in X$. $X \times Y$ is not the same as $Z$; they only, effectively, contain equal elements.  He goes on:     If $\{X_i\}$ is a family of sets ($i \in I$), the Cartesian product of the family is, by definition, the set of all families $\{x_i\}$ with $x_i \in X_i$ for each $i$ in $I$.   How I understand it: $\{X_i\}$ contains, as elements, several sets, each single one a set $X$ identified by an index/element of $I$. Then, we go over all indeces $i$ in $I$, take from each set $X$, namely $X_i$, an element $x$ and add it to current set of the newly created sets/families. So, the lower index $i$ does not refer to an element of $X_i$ indexed by $i$, but rather calls all elements of $X_i$ $x_i$. We end up with a set, defined as Cartesian product, of sets, in which each element (set) contains ordered elements, each from a different $X_i$. How does the ordering occur?  $\prod_i X_i$, the Cartesian product, becomes equal to $X^I$, if all $X_i$ are the same set $X$. $X^I$ means all possible combinations of mapping from $I$ to $X$. We randomly take an elements $x$ from some $X_i$, put it in a set/family with other randomly obtained $x$, put the entire set into a set denoted $X^I$, and continue doing so, till we have no more options. Then, each set in the new $X^I$ contains a mapping of some $i$ to some $x$. Since we exhausted the options, they are indeed all combinations.  Where did I go wrong? Someone needs to spoonfeed my the answer; I am completely confused by now.