I'm studying for my final for Statistics, and I want to understand literally every problem in my textbook (at least in the first 7 chapters).  One of the problems asks to show that ${n}\choose {k}$ $=$${n-1}\choose {k}$ $+$ ${n-1}\choose {k-1}$  I have the solution manual, but the steps it gives appears to skip a few important steps or leave out important clarifications.  Can someone please guide me step by step through this proof?
I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas?
Moderator flag notifications should reach mobile I spend a great deal of time surfing SE sites from my phone during commutes and other down times. The global inbox is quite effective at reaching me even for chat pings. Quite frankly the mobile site is very well done!  However as a community moderator, there is a whole class of information not getting through. There are no notifications for moderator flags! I can switch to the full site and see these but that is inconvenient for browsing, so I only end up checking for flags periodically.  Ideally the moderator tools should all be available in mobile templates, but as a stand-by it's reasonable to switch to the full view to handle flags, but flags would get addressed in a much more timely manner if I knew about them. As it stands I often come and go from the site without even knowing there are outstanding flags.  A simple flag counter added to the header next to the notification counter and linked to the tools page (which automatically opens the full version anyway right now) would fix this. I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that one can arrange all the numbers from $1$ to $\color{red}{305}$ in a row such that the sum of every two adjacent numbers is a perfect cube. $$256,87,129, 214, 298, 45, 171, 172, 44, 299, 213, 130, 86, 257, 255,$$ $$88, 128, 215, 297, 46, 170, 173, 43, 300, 212, 131, 85, 258, 254, 89, 127, 216, 296,$$ $$ 47, 169, 174, 42, 301, 211, 132, 84, 259, 253, 90, 126, 217, 295, 48, 168, 175, 41, 302, $$ $$210, 133, 83, 260, 252, 91, 125, 218, 294, 49, 167, 176, 40, 303, 209, 134, 82, 261, 251,$$ $$ 92, 33, 183, 160, 56, 287, 225, 118, 98, 245, 267, 76, 140, 203, 13, 14, 202, 141, 75, 268,$$ $$ 244, 99, 26, 190, 153, 63, 280, 232, 111, 105, 238, 274, 69, 147, 196, 20, 7, 1, 124, 219,$$ $$ 293, 50, 166, 177, 39, 304, 208, 135, 81, 262, 250, 93, 32, 184, 159, 57, 286, 226, 117, 8,$$ $$ 19, 197, 146, 70, 273, 239, 104, 112, 231, 281, 62, 154, 189, 27, 37, 179, 164, 52, 291, 221,$$ $$ 122, 3, 5, 22, 194, 149, 67, 276, 236, 107, 109, 234, 278, 65, 151, 192, 24, 101, 242, 270,$$ $$ 73, 143, 200, 16, 11, 205, 138, 78, 265, 247, 96, 120, 223, 289, 54, 162, 181, 35, 29, 187,$$ $$156, 60, 283, 229, 114, 102, 241, 271, 72, 144, 199, 17, 108, 235, 277, 66, 150, 193, 23,$$ $$ 4, 121, 222, 290, 53, 163, 180, 36, 28, 188, 155, 61, 282, 230, 113, 103, 240, 272, 71, 145,$$ $$ 198, 18, 9, 116, 227, 285, 58, 158, 185, 31, 94, 249, 263, 80, 136, 207, 305, 38, 178, 165,$$ $$ 51, 292, 220, 123, 2, 6, 21, 195, 148, 68, 275, 237, 106, 110, 233, 279, 64, 152, 191, 25,$$ $$100, 243, 269, 74, 142, 201, 15, 12, 204, 139, 77, 266, 246, 97, 119, 224, 288, 55, 161,$$ $$ 182, 34, 30, 186, 157, 59, 284, 228, 115, 10, 206, 137, 79, 264, 248, 95$$ Here, I have a question.  Question : Does there exist at least one positive integer $n\ge 2$ satisfying the following condition for each $N\ge 2\in\mathbb N$? Condition : One can arrange all the numbers from $1$ to $n$ in a row such that the sum of every two adjacent numbers is of the form $m^N$ for some $m\in\mathbb N$.  Added : I crossposted to . Suppose we have a sample of size n= 9, where the sum equals 27, and the sum of the squaredvalues equals 113. Sample mean is 27/9=3 and mean of squared values is 113/9 = 12.56 Can we then say that the sample variance $S^2 = E(\overline X^2) - E(\overline X)^2 = 12.55 - 9 = 3.56$ ? It seems that in the words Englishman, Frenchman, and Scotsman, the ‑man part is pronounced /mən/ (just like in Roman).  Whereas in snowman, the ‑man part is pronounced /mæn/ (just like in no man).  Why is it that when ‑man is appended to snow‑ to make snowman, the pronunciation of the man part doesn’t change?    Wouldn’t it logically follow for it to be pronounced /ˈsnowmən/ not /ˈsnowˌmæn/? Beautiful table samples I'm interested in learning about some nice-looking table samples. Colors can make a huge difference, and just providing some color to headers, background, or borders can dramatically change presentation. Any samples you're proud to share?  (BTW here's a sample I like, but unfortunately, not TeX...) Finder: keep sort-by per-directory How can I dynamically load the functions of java.lang.Math How to display Toast in Android? What is the difference between a "link function" and a "canonical link function" for GLM What's the difference between terms 'link function' and 'canonical link function'? Also, are there any (theoretical) advantages of using one over the other?  For example, a binary response variable can be modeled using many link functions such as logit, probit, etc. But, logit here is considered the "canonical" link function. Expression Versus Statement I'm asking with regards to c#, but I assume its the same in most other languages.  Does anyone have a good definition of expressions and statements and what the differences are? I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? Any help would be appreciated.  If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? How to read a text file reversely with iterator in C# I need to process a large file, around 400K lines and 200 M. But sometimes I have to process from bottom up. How can I use iterator (yield return) here? Basically I don't like to load everything in memory. I know it is more efficient to use iterator in .NET. Elements order in a "for (… in …)" loop Does the "for…in" loop in Javascript loop through the hashtables/elements in the order they are declared? Is there a browser which doesn't do it in order? The object I wish to use will be declared once and will never be modified.  Suppose I have:  var myObject = { A: "Hello", B: "World" };   And I further use them in:  for (var item in myObject) alert(item + " : " + myObject[item]);   Can I expect 'A : "Hello"' to always come before 'B : "World"' in most decent browsers? Option clash for package xcolor I'm writing a report, and have a rather large table in my appendix. I'd like to emphasize some specific values by coloring some single cells. I'm trying to do it with xcolor. But receiving the error:  ! LaTeX Error: Option clash for package xcolor. See the LaTeX manual or LaTeX Companion for explanation. Type H &lt;return&gt; for immediate help. ... l.17 The package xcolor has already been loaded with options: [] There has now been an attempt to load it with options [table]   My preamble looks as follows  \documentclass[a4paper,oneside,article]{memoir} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[danish]{babel} \usepackage{lmodern} \usepackage{graphicx}     \graphicspath{{Figures/}} \usepackage{fullpage} \usepackage{modroman} \usepackage{float} \usepackage{caption} \usepackage[numbers]{natbib} \usepackage{url} \usepackage{pgfplots} \usepackage{mathtools} \usepackage[table]{xcolor}   And take out of my table  \begin{table}  \centering   \begin{tabular}{|c|c|c|}        \hline    \cellcolor{green!25}D &amp; x &amp; D/$L_0$ \\        \hline    65 &amp; 2600 &amp; 0,417\\        \hline    60 &amp; 2400 &amp; 0,385\\   \end{tabular}  \caption{Bestemmelse af bølgehøjde}  \label{tab:Q1} \end{table}   Read some other questions asking more or less the same. And the answer to them seems to be a "collision" between packages. I have no idea how to find the problem. So my question is:   Where is the "collision"? How on earth would I have found it without help ? Is there a Unix command to get the absolute (and canonicalized) path from a relative path which may contain symbolic links? What I want is to create a bootable pen drive that shows GRUB menu and I can select any of the several bootable systems from there to install/run. I have several ISO images that I want to install in the USB like Ubuntu, Fedora, System Rescue CD, Windows XP and Windows 7 (The windows versions are optional, mostly am looking for a multi installer for Linux distros, specially Ubuntu version)  The default Startup Disk Creator only works with Linux Images and only 1 at a time (At least for the mean time. If it allows more systems and multiple ISOs it would be awesome)  What method exists that I can use step by step to create a bootable USB Drive with multiple images than can be booted from the USB and selected from GRUB. It must support Windows Images and Linux Images.  Alternatively, is there also a way to have multiple versions of the Ubuntu installation ISO on same USB Drive, such as, when booting from it, I get to choose what Ubuntu version and architecture to install, for example:     BOOT MENU   Ubuntu 12.04 32 Bit   Ubuntu 12.04 64 Bit   Ubuntu 12.10 32 Bit   Ubuntu 12.10 64 Bit   UPDATE: Wanted to add that the English version for the post mentioned by daithib8 is here: How to show a group is cyclic? One question asking if $\mathbb{Z}^*_{21}$ is cyclic.  I know that the cyclic group must have a generator which can generate all of the elements within the group.  But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine if a group is a cyclic group? In Harry Potter, Do Spells Travel in a Straight Line? In the Harry Potter world, when a magical person casts a spell, sometimes they target a specific object, like when Harry has a wand only and needs his broom, he casts "Accio Firebolt" and his broom comes to him.  But a lot of the time people point their wands just in the direction of someone to cast a spell.  One example would be in a duel.  When they do this, are they casting a spell on a particular person or at that person?  It seems, from what I remember of the movies, that some spells travel in a straight line while others specifically target a person or object.  Do any spells travel in one direction, like a laser beam?  What kind of spells are directional (instead of focusing on a particular person or object)?  And can directional spells be deflected or blocked by something like a mirror or another physical object?  (By "directional," I could have said linear, but didn't want to start sounding mathematical or scientific.  In other words, do they travel in one direction, like a laser beam would, or will they change direction to hit the intended target?) What is the use of the yield keyword in Python? What does it do? For example, I'm trying to understand this code1: def _get_child_candidates(self, distance, min_dist, max_dist):     if self._leftchild and distance - max_dist &lt; self._median:         yield self._leftchild     if self._rightchild and distance + max_dist &gt;= self._median:         yield self._rightchild    And this is the caller: result, candidates = [], [self] while candidates:     node = candidates.pop()     distance = node._get_dist(obj)     if distance &lt;= max_dist and distance &gt;= min_dist:         result.extend(node._values)     candidates.extend(node._get_child_candidates(distance, min_dist, max_dist)) return result  What happens when the method _get_child_candidates is called? Is a list returned? A single element? Is it called again? When will subsequent calls stop?   1. This piece of code was written by Jochen Schulz (jrschulz), who made a great Python library for metric spaces. This is the link to the complete source: [Module mspace][1].