The Summation Identity is: $\dbinom{n}{r}+\dbinom{n}{r+1}=\dbinom{n+1}{r+1}$. Is there a way to prove the hockey stick identity with this?  Hockey Stick Identity: $\dbinom{n}{r}+\dbinom{n+1}{r}+\dbinom{n+2}{r} + \cdots + \dbinom{f}{r}=\dbinom{f+1}{r+1}$
After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
Different mechanics of \hyperlink vs. \hyperref I am fiddling around with the hyperref package to build a hypertext structure into my pdf. At first glance, hyperref provides two similar macros to put links between parts of my document: \hyperlink{target}{text} and \hyperref[target]{text}. You might notice their slightly different syntax. What really interests me is their different working mechanics. I will illustrate this by the following example: \documentclass{article}  \usepackage{pdfpages} \usepackage[]{hyperref} \usepackage{mwe}  \begin{document}  \section{Section Heading} \label{sectionheading}  \includepdf[link=true, linkname=pdfpage]{example-image}   Link to section works with \hyperref[sectionheading]{hyperref} and with \hyperlink{sectionheading}{hyperlink} (but with slightly different targets on the page).  Link to pdf works with \hyperlink{pdfpage.1}{hyperlink} but not with \hyperref[pdfpage.1]{hyperref} (and not this way, either: \hyperref[pdfpage]{hyperref}).  \end{document}  To link to the labeled section heading, I may use both commands. While \hyperref directs my pdf viewer directly to the linked section, \hyperlink directs me somewhere slightly different, so I can see more context. To link to the \includepdf of pdfpages however, I may only use \hyperlink. \hyperref does not work at all. I suspect this is because of a subtle difference in their working mechanisms. While checking the aux file, I found a line \newlabel{sectionheading}{{1}{1}{Section Heading}{section.1}{}} which \hyperref seems to use as a target. However, I did not find any similar target for the included pdf. My question: I suspect, I understand the workings of \label and \ref very well. \hyperref seems to work quite similarly with the added clickability. Could you please explain to me how \hyperlink does work differently and how it is implemented? E.g. what mechanics explain the difference when clicked on the link and where and how are the link targets stored in the first place? Fix SD card that cannot be formatted Are there number systems corresponding to higher cardinalities than the real numbers? How to Understand the Definition of Cardinal Exponentiation I'm having trouble understanding the definition of cardinal exponentiation.  Let's start with the definitions / claims I've been given:  For any finite sets $A,B$, such that if $|A|=a$ and $|B|=b\neq 0$ then the number of functions from $A$ to $B$ is $b^a$. ( I'm guessing this is only used when both sets are finite ).  For any sets $A,B$, $B^A$ is the set of functions from $A$ to $B$.  For every set $A$, $|P(A)|=|\left \{ 0,1 \right \}^A|$  Now for the definition and why I don't understand it:  Let $a,b$ be cardinal numbers.  Let $A,B$ be sets such that $|A|=a, |B|=b$.  Then $|B^A|=b^a$.  How analagous to normal exponentiation is this and where does in differ when we introduce cardinals like $\aleph_0$ or $C$?  Is $2^2=4$ and is $\frak c^2=c\cdot c=c\times c$? I've had the ability to review suggested edits for a while now, however I don't often do so because the list is usually full of questions about languages and technologies I'm not familiar with or not interested in, and can sometimes be hundreds of questions long.  I would like to help out more, but most of my rep has been gained by questions within a specific technology, so I only feel comfortable reviewing edits within that tag.   Can you allow us to filter suggested edits by tag?  Edit:   Actually, can you let us filter the entire Review section by tag, and not just Suggested Edits?  I would find it much more interesting to review questions in tags I'm familiar with and can  understand the question content of, and it would give me a bit of pride to help maintain "my" section of SO. How can I make a hexagonal grill? Which countries require a 24-hour transit visa in China? I'm transiting through China in Kunming, China.  I'm a South African national and do NOT fall under the 72-hour visa free program. However on [their official (travelchina) website][1] it says:     Nationals of most countries are eligible for the 24-hour direct   transit, while a visa is needed for the passengers transiting on   Afghanistan, Azerbaijan, Iran, Iraq, Kazakhstan, Kyrgyzstan, Malaysia,   Nigeria, Oman, Pakistan, Saudi Arabia, Sri Lanka, Thailand, Turkey,   United Arab Emirates and Yemen passports.   I'm sorry if I'm missing this, but the above is rather badly worded for me does "nationals of most countries" include ALL countries except those mentioned in the list shown?  Also again sorry if I'm missing this but again it is rather badly worded for me. I will fly via Thailand to China and then via China to Canada with an 8-hour layover in China. I see Thailand is mentioned on the list. I have a Thailand residency permit where I currently reside... Would that affect me at all?  Also, any idea if I can exit the airport in China and come back under the 24-hour visa exemption (should I be exempt)?  Many thanks. What is the name of a small unluxurious restaurant? Is there a name for this kind of restaurant? Searching Google, I can't find any synonym of restaurant. I've tried to search for images of unluxurious restaurant, small restaurant or mundane restaurant, with no satisfactory result. The restaurant I'm looking for is small, not so clean and definitely not a place that high class people go to eat.  It can be a house:    Or lie on a side of the street: Filtered Review Queue doesn't show any audits What is polarisation, spin, helicity, chirality and parity? Polarisation, spin, helicity, chirality and parity keep confusing me. They seem to be related, but exactly how they are related is unclear to me. Can someone maybe give a short overview about what these quantities mean and how they are related? What are the values for e.g. basic particles like electron or photon? Idiomatic expression for a difficult choice This has cropped up several times in the past couple of months, and I've been struggling to find  a fitting word to describe this phenomenon. I'll describe it:  You have two choices(no, it's not Hobson's choice). Both of them are desirable, yet both of them each have a tiny part which is most undesirable, such as a tough condition, or a demanding clause. You have difficulty deciding which one to pick.  Whenever something like that comes up, I think of the expression "Can't burn a candle at both ends", but I'm not looking for that.  I'm looking for an expression that describes a difficult choice. Any suggestions?  EDIT: It's idiomatic. I'm quite sure it is. It's an expression of some sort. And no, it isn't "dilemma" Connect to SQL Server with Windows Authentication in a different domain Read-only file system Big Greeks and commutation In my , I provided the following, problematic,  wording (especially bold italic), and I need help to better understand this issue so I can fix my answer:1          The thing is the books. (Reduced form of Sentence 2.a)         Fundamentally, Sentence 2.a (the so-called "correct" answer), is grammatically defective. Recall there is another grammatical rule: the subject and subject's complement should match in number.    A clarification is needed here. I am not suggesting here, in this EL&amp;U question, that there is any such singular grammatical rule as indicated in the quote above.  The quote continues as follows:     The reduced sentence makes the disagreement between the subject's and complement's plurality obvious. What is one to do? Language is linear and we know "The thing is X" is better than "The thing are X", so we go with the former, which is the subject-verb agreement rule. But what is "the books" then? It must be thought of as a collective noun, even if it doesn't look or feel like one. Forcing something to be a collective noun is related to the idea of notional agreement.   Others noted that grammatically defective is too strongly worded.  There is no such "grammatical rule" for subject/complement agreement.  Please help me understand/improve upon my line of thought here.  I'm stating that Sentence 2.1 presents a fundamental "conflict" or "error" (however one might define error) at some level (morphological, syntax, grammatical, semantic):     The thing is the books.  (Or, "The thing is the objects.")   Based on the "obvious" morphosyntactic plurality conflict, the above sentence feels so wrong that I think many native speakers would think it "simply" grammatically incorrect. However, @F.E. gives counter examples:     And here are some more grammatical examples, where the number of subject and predicative complement don't agree: "They were a problem to us all", "That so-called work of art is simply four pieces of driftwood glued together" (CGEL, page 254-5). As CGEL says: "What is required is semantic compatibility, not syntactic agreement . . .". There are more examples on page 512: "The only thing we need now is some new curtains", "The major asset of the team is its world-class opening bowlers", "Our neighbors are a nuisance", "This gadget is five different tools in one".   I think all the examples given above can be explained as notional agreement.    Question 1: If we explain this phenomena in terms of notional vs syntactic agreement, where does "grammatically correct/incorrect" fit in?  Question 2: Can a simple sentence such as "The thing is the objects." be grammatically incorrect while more complex sentences such as "The thing is four pieces of driftwood glued together." be grammatically correct?  (The answer, in my mind, must be no; this is my conundrum.)  Question 3: Would it be better to say that Sentence 2.a has a low level of (linguistic) ?   Question 4: How about a low level of  and that it's semantically difficult to be (linguistically) ?    Question 5: Do we simply draw a hard line and say "The thing is the objects." is 100% grammatically correct?  I didn't particularly like this option based on intuitive notions of "grammatically correct".    1. Please help me and be kind; I'm trying to improve my understanding. I'm looking for cool, objective advice on how to look at this.  I numbered my questions so they can be easily referenced, if desired.  My preference is the final answer would sufficiently answer all the questions, but of course that can be done explicitly or implicitly. How to use `ybar interval` without skipping the last value I have the following code  \documentclass[tikz]{standalone} \usepackage{pgfplots} \usepackage{pgfplotstable} \usepackage{filecontents}  \begin{filecontents}{testdata.dat}   -3.316988309367815e-07 0.07000000000000001   -1.948815911147526e-08 0.13   7.15864175355987e-08 0   1.799432483450127e-07 0   0.0002850606647384129 0   0.0009429559686380238 0.08999999999999997   0.00157188462064142 0   0.005259918700865051 0   0.01739392641273605 0.18   0.06208364670743995 0.55   68.69485148165926 6.970000000000001   31.217611224923 79.12   0 12.89 \end{filecontents}  \pgfplotsset{compat=newest} \pgfplotstableread{testdata.dat}\datatable  \begin{document} \begin{tikzpicture} \begin{axis}[   ybar interval,   width=10cm,   height=5cm,   enlargelimits=false,    xtick=data,   ymin=0, ymax=100,   xticklabels={0 -- 0.063,0.063 -- 0.125,0.125 -- 0.25,0.25 -- 0.5,0.5 -- 1,                1 -- 2,2 -- 4,4 -- 5.6,5.6 -- 8,                8 -- 11.2,11.2 -- 16,16 -- 22.4,22.4 -- 31.5},   x tick label style={rotate=60,anchor=east}   ]   \addplot table[x expr=\coordindex,y=0]{\datatable};   \addplot table[x expr=\coordindex,y=1]{\datatable};  \end{axis} \end{tikzpicture} \end{document}   and the output looks like this:    But the command ybar interval does skip the last value. It should look like:    I prefer to have the last value to show up in the interval 22.4 -- 31.5. How do I achieve this without inserting a row of zeros to my table? Find remainder of division of $x^3$ by $x^2-x+1$ I am stuck at my exam practice here.      The remainder of the division of $x^3$ by $x^2-x+1$ is ..... and that of $x^{2007}$ by $x^2-x+1$ is .....   I tried the polynomial remainder theorem but I am not sure if I did it correctly.  By factor theorem definition, provided by ,     the remainder of the division of a polynomial $f(x)$ by a linear polynomial  $x-r$ is equal to $f(r)$.   So I attempted to find $r$ by factorizing $x^2-x+1$ first but I got the complex form $x=\frac{1\pm\sqrt{3}i}{2}=r$.   $f(r)$ is then $(\frac{1+\sqrt{3}i}{2})^3$ or $(\frac{1-\sqrt{3}i}{2})^3$ which do not sound right.  However, the answer key provided is $-1$ for the first question and also $-1$ for the second one. Please help. Using zypper, how can I remove all packages installed from a specific repository? During the beta of Windows 10 it was hard know what version you were running unless it had it posted to the desktop. Once that wasn't there - how do you tell what version / build you are running?   This will become more of an issue as Microsoft starts releasing more builds with the new update mechanism. Global setting of hyphenation for words containing dashes