I've been trying to think up of two combinatorial proofs but I'm confused on the first and just completely stuck on where to begin for the second.   $$\sum_{k=-m}^n \binom{m+k}{r}\binom{n-k}{s} = \binom{m+n+1}{r+s+1}$$ For the RHS, I understand that it is basically looking at the number of $(r+s+1)$ subsets that can be formed from $(m+n+1)$. On expanding the LHS to yield $\binom{0}{r}\binom{n+m}{s}+\binom{1}{r}\binom{n+m-1}{s}...\binom{m}{r}\binom{n}{s}+\binom{m+1}{r}\binom{n-1}{s}+...\binom{m+n}{r}\binom{0}{s}$, I see that it is basically adding together the various ways of drawing a constant $r$ and $s$ from two changing populations that sum up to $m+n$. However, I don't actually see where the $+1$ comes from. Is there anything wrong with my intepretation of the LHS? $$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$ I know that RHS is counting the number of subsets of 3 that can be formed from $n$. However, I do not have any idea of how to be interpreting the LHS.
Restrictions on the variables for this identity are as follows: $a \geq m \geq 0$ and $n \geq b \geq 0$.  I have not gotten far with this, but these are my very rudimentary thoughts: We need to define a special element, somehow, and then carve up the $m + n$ remaining elements to be chosen out of the remaining pool of $a + b$ elements by writing $a + b = a - i + b + i$. From the $a - i$ pile, we choose $m$ things. From the $b + i$ pile, we choose $n$ things. My problem is, what is going on with this special element? How is it being defined so as to split things in this precise way?  A hint would be great for the moment.
Domain of ProbabilityDistributions Labelling figures in the tikz package Consider the following piece of code, which uses the tikz-feynman package:  \usepackage{tikz-feynman, contour} \tikzfeynmanset{compat=1.1.0}  \begin{document} \begin{center}     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (c) -- [fermion] (b)                 -- [fermion] (d)                 -- [fermion] (a)                 -- (c),         };     \end{feynman}     \end{tikzpicture}     %     \feynmandiagram[baseline={(current bounding box.center)}, vertical'=a to b,] {         {   [edges={fermion}]             a -- b [label=0:\(\mu_{3}\)]             -- c [label=-180:\(\mu_{4}\)]             -- d [label=-180:\(\mu_{1}\)]             -- a [label=0:\(\mu_{2}\)],         },         {   [edges={photon}]             a -- ap [particle=\(p_{2}\)],             b -- bp [particle=\(p_{3}\)],             d -- dp [particle=\(p_{1}\)],             c -- cp [particle=\(p_{4}\)],         },     };     %     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (a) -- [anti fermion] (b)                 -- [anti fermion] (d)                 -- [anti fermion] (c)                 -- (a),         };     \end{feynman}     \end{tikzpicture}     \end{center} \end{document}   This piece of code draws the following figure:    I would like to label each figure. I do not want a caption, but just a label beneath each diagram. The label should say Diagram I, Diagram II, Diagram III.  How do I implement this into my code? In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?  I found PMF of $X$ and it is, if $k &gt; n+1$  $$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$  How could I evaluate $E(X)$?  Edit: I'm looking for an explicit form of $E(X)$ Describe the set of points on the complex plane... What does a confidence interval with a negative endpoint mean? Basically, I've designed a booklet on Photoshop CC. It's an informational booklet so it's quite vibrant with lots of colours and pictures in it.   I've finished it and exported all the files into PDF format. Upon doing this, the PDF's are much "duller" for use of a better word. Now I'm not a designer and the whole thing was a learning process but now I'm reading about CMYK and RGB colour models and conversions etc. I designed the whole thing RGB (because CMYK looked duller obviously. The PDF's aren't disastrous, red is still red, black is still black but it's nowhere near as visually appealing anymore.   Is this just because of the monitor and the models each program uses or when I send my brochure to be printed, am I doomed for a much duller booklet? There's obviously a way around it otherwise all print would be dull. I'm sorry if this is a broad question but can someone explain to me how to interpret these regression diagnostic plots? I understand the Normal Q-Q show's how normal the spread of the data is, but the other three I am confused by. For the Residuals vs Leverage, what does it mean for a point to be outside of the bounds? What does it mean if the red lines are parabolic and curved like in the first two? Why do we care about residuals, and what do they tell us? I've searched the web multiple times for some guide to interpret these and I can't get any straight forward answers. Any help is appreciated, thank you. Layers with same coordinates not lining up Store update, insert, or delete statement affected an unexpected number of rows (0) EntityFramework Does SQLDeveloper support executing scripts? Can I still use the older Airport Utility 5.5.3 if I install the 6.0 version of Airport Utility? Difference between a Structure and a Union I have a 2&nbsp;GB Kingston  card which is about three years old. I put it in a reader today in my Windows Vista computer, wrote a 32&nbsp;MB file onto it, safely removed it, and then tried to read it elsewhere. Nothing. Putting it back in the Windows Vista computer it now says:     You need to format the disk in drive F: before you can use it.   What should I do? I have access to many computers and OSes if your recommendations need that. I would be very sad if I lost all the contents of the card. Most of the data is backed up, but there are a few things that aren't. :(  Doing a   # dd if=/dev/sdg of=~/tmp/sd.bin   gives me a 2&nbsp;GB file, and  the file it seems like lots of my data is still there. How can I put it back together? Word: how to show hidden ribbon tabs? How can I control which tabs are available to be opened in the MS Word 2010 ribbon?  Under Word Options > Customize Ribbon > Main Tabs there are several groups checked (Blog Post, Insert (Blog Post), Outlining, Background Removal), but they don't appear on the ribbon even though there is space for them. How is one supposed to turn them on or get to them?  The one in particular I'm after today is the Blog Post tab, which I know I can get to in a round about way via File > New > Blog Post Template. However I don't want a new post, rather an old one, which means File > New > Blog Post Template > Open existing > [select from list] > [close new blog post window] > [slap No I don't to save darn it]. Extra work for nothing is irritating; how do I get Blog Post (or Outlining or ...) to show up in that nice empty space next to AddIns? Rendered image is just grey I am having some issues when rendering an image, my rendering screen is turning into grey even when making a full render. What am I doing wrong? Imported 3D scan displays texture in editor, but renders without texture I am new to blender and I have a 3D scan of a stone tablet (which I am unfortunately not able to share, so I will try to be as detailed as possible) which was imported as OBJ and MTL files.  The editor displays the texture and mesh perfectly, but when I render the scene (pressing F12) the texture is not there.  I am trying to use blender to reduce the file size of the tablet for use in a three.js visualizer for a website.  I have tried exporting the file as a JSON file (as well as several other file formats to the same effect) for use in my three.js webapp and thought it was an issue with three.js, but after seeing that blender did not render the tablet with the texture I now believe it is an issue with my blender configuration. I am unable to find within blender or online any way to check what current resources I have displayed and what options I have with them.  I have not applied a mesh or texture to the object, but when I do so it renders with the generated texture applied in the default pink.  I am confused as to where the disconnect is between the editor and the renderer. I am still figuring out what is important and where to find all the relevant information, so any help on that regard would be very much appreciated.  Here is what blender looks like for me with the sensitive parts censored (can't be too careful these days):   I have looked at this question:  And there is no material overwriting the texture - I have not assigned a material to the tablet at all. EDIT: In regards to the duplicate answer, I looked at that one as well and the issue I have is that it doesn't have an actual texture file that I can just put on it.  The OBJ MTL files come with their own file that can't be applied directly to the model.  Blender read the OBJ MTL files and recreated the object inside blender algorithmically - and now I need to know how to get blender to tell me where that texture is and how I can get it to show up in the render.  The texture is not on my hard drive, but entirely within blender which is the issue I am having. Can someone please elucidate the difference between "many" and "many a"? In what context of usage should we add an extra "a" beside the word "many"?  For example:         Many times, I had seen that . . . .   Many a times, I had seen that . . . .      I do not know if the example is accurate enough to support my question, but I would like to know the difference between how they are used. Why don't std::algorithms work directly on containers as well? I am looking for a pipe symbol that can be made to vary in height automatically.  One of the common ways of writing "such that" in mathematics is with the pipe symbol (|). The pipe can be represented in LaTeX directly as the character | or as \vert, but neither of these expands vertically in math presentation mode. I can adjust it manually this way:  \[\mathcal{H}=\left\lbrace h(x)=\sum_{i=0}^{k-1}a_{i}x_{i}\mod m\ \Bigg\vert      \ a_{i}\in\mathbb{Z}_{m^{k}}\right\rbrace\]     but I am looking for a true variable-sized option, parallel to \lbrace etc.  The amsmath package provides \lvert and \rvert, but I do not find them expanding as expected, and the amsmath documentation specifies that they should be used as binary delimiters (one on each side of something else), rather than alone. I've read  about how some publishers (e.g., Springer) allows authors to submit to arXiv before journal submission, but they make authors wait for a year after acceptance to submit to arXiv.   My question is: Is it okay to submit to arXiv during the time when the paper is under consideration by the journal, for example after it has been "revised and resubmitted" but before final decision?