How to remove Ubuntu 15.10 from a multiboot configuration?
How do I completely remove Ubuntu?
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$  I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$  But I don't think it can lead me anywhere. I would like the proof to be as simple as possible. I keep seeing places refer to the "multiverse" repository as a place I can get software, how can I enable this repository? Please specify how to do this graphically and via command line. Number of possible marriage arrangements if Marriage Condition is satisfied Suppose I have a set of $n$ boys and each subset of $1\leqslant k \leqslant n$ boys knows at least $k$ girls then this means that the set of boys $B$ satisfies the marriage condition.  I am doing an exercise from a Graph Theory book and it says we can show by induction on $n$ that the number of possible marriage arrangements is $\geqslant t!$ if every individual boy knows at least $t \leqslant n$ girls or there are $\geqslant t!/(t-n)!$ possible marriages if each individual boy knows at least $t&gt;n$ girls.   I am struggling to see how the induction hypothesis can be implemented to prove the $n+1$ case. When you swap from the $n$ to $n+1$ case the conditions change and I can't see the link between them.  Would anyone be able to point me in the right direction? [Probability]need help to understand the following expression So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively.  assume c is a constant.  I am having huge trouble to understand the integration of the following expression.  $P(Y&lt;c/u(X))$  $=\int_{t}^{\infty}f_X(x)\int_{0}^{c/u(x)}f_Y(y)dydx +\int_{0}^{t}f_X(x)\int_{c/u(x)}^{\infty}f_Y(y)dydx$  where t is the cross-point that $u(x)$ change sign    here $c/u(x)$ is given by the plot below, t is the point crossing the zero:   Confusion:   I don't understand the second integration of the second term "$\int_{c/u(x)}^{\infty}f_Y(y)dydx$", this isn't right because Y an X only defined for y>0, and x>0. So it's the first quadrant in this plot. Applying for a Canadian ETA after previous visits without one Is there any Stack Exchange site where I can ask others to recommend me good web sites for solving specific problems?  For example I would like to know if there is any URL shortening website that can allow me to define the url.  For example instead of http://bit.ly/1gX2dD0 I would like it to be http://bit.ly/superflow - if "superflow" is not available then I will invent another word.  I can imagine that such services are paid but I don't need the link to last more than one day, and maybe there is such a website. How do I change my update server in Ubuntu 18.04? Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the three conditions happens. I am reading George Gamow's book  and a certain assertion Gamow makes seems a little startling considering what I already know of the subject of infinite cardinality.     But the number of all geometrical points, though larger than the number of integer and fractional numbers, is not the largest one known to mathematicians. In fact it was found the variety of all possible curves, including those of the most unusual shapes, has a larger membership than the collection of all geometric points, and thus has to be described by the third number in the infinite sequence. [Gamow, 22]   He later does explicitly say the number of all possible curves is $\aleph_2$. But nowhere else can I find reference to this being true, nor does he provide any sort of reasoning to its validity.  My question is how can you prove the set of all possible curves is of cardinality $\aleph_2$ and not $\aleph_1$ or $\aleph_0$? RSA encryption with private key and decryption with a public key Is there a c# equivalent of Java ConcurrentHashMap Bibliography with page numbers In GKP's Concrete Mathematics, there is a useful and interesting feature I would like to see in more books. The bibliography entries include:   a - the page numbers where the book or article is cited in the document. b - a few lines of review of the cited work.   It is a kind of mix between bibliography, index and review.  Are there stable LaTeX packages/BibTeX styles to do (a) and (b) ?  EDIT: I wrote (b) initially as "comments about the work cited". It could be understood as "gathering comments made in the book about the work and reproducing them in the bibliography".  This is not what I meant but this could be something useful for others. "He was telling me that he is going..." vs. "He told me that he is going..." Accepted way to abbreviate cities that have "Saint" in their name My first question here, so I'm not sure if this OT or not, so please bare with me.  Background: I don't live in US. I'm working with some data that I'm pulling from . Sometimes, I'm getting a city name as St. John and sometimes as Saint Ann.   So my question is, is there an accepted way on how people write a US city names which starts with Saint or is this just some random thing (or just specific to AdWords maybe)?  I'm asking this, because if for example I'll get both St. John and Saint John in the same file, should I consider them the same or should I be suspicious? Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? This is a question of the book Linear Algebra of Kenneth Hoffman.  Describe explicitly all inner products on $\mathbb{R}$ and on $\mathbb{C}$.  I think that if $&lt;,&gt;$ is one inner product on $\mathbb{R}$ (or $\mathbb{C}$) then $&lt; ,&gt;=K[,]$, where $[,]$ is the standard inner product. Is that correct? Why cast an unused function parameter value to void? Is it okay to keep flour in the freezer? Years ago I noticed I was getting little bugs in the flour that I kept in the pantry.  To prevent this, I started keeping my flour in the freezer.  I don't do a lot of baking, but I was wondering what effects this might have on anything I do bake.  Do baked goods turn out any differently when flour is kept in the freezer vs flour kept at room temperature?  Also, if it's okay to freeze flour, how long will it last this way? I have a bunch of magnets (one of those game-board thingies) given to me when I was a school-going lad over 20 years ago, and the magnets feel just as strong as it was the day it was given.   As a corollary to this question , is there a way to determine how long a permanent magnet will remain a magnet?   Addendum: Would two magnets remain a magnet for a shorter duration if they were glued in close proximity with like poles facing each other? I would like to create an alias to rm command in order to have a confirmation message after executing this command. So I am creating an alias like this alias rm='rm -i'. But as far as I know this is a temporary alias and it lives until you close the terminal.  As it is explained  to save alias permanently I need to execute ~/.bash_aliases or ~/.bashrc commands in terminal and add my alias there. But when I execute ~/.bashrc I get following error message :  bash: /home/bakhtiyor/.bashrc: Permission denied   When I run ~/.bash_aliases I get another error message like this:  bash: /home/bakhtiyor/.bash_aliases: File or directory doesn't exist.   What is the actual problem and how can I solve it?