The closed form for the number of 2-element subsets of the power set of a set with fixed overal size I am looking for a closed form for the sum below  $$ \sum_{k=0}^{r} \frac{1}{2-\delta_{k, r-k}} \binom{n}{k} \binom{n}{r-k} $$  where $\delta_{k, r-k}$ is the Kronecker delta. I'd appreciate any helps in finding a closed-form solution for this sum.   To my understanding, this is basically equivalent to finding a closed-form solution to the following problem:  Given a set $S$ with size $|S|=n$, what is the number of two-element subsets of $\mathcal{P}(S)$, $\{A, B\} \in \mathcal{P}(S)$ such that $|A|+|B| = r$  for a fixed $r$.  I tried to use Wolfram Alpha, but it is given me this for the case where we count ordered pairs $(A, B)$ instead of sets (i.e. the sum if we ignore $\frac{1}{2-\delta_{k, r-k}}$), which contains factorial of a negative integer(!!) (and I don't know how to specify in Wolfram that $n&gt;r&gt;0$ and both are integers):  $$ \frac{(-1)^r (-2n+r-1)!}{(-2n-1)!r!} $$
How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$? How can we prove that     $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$   (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
Wikipedia onebox is dropping random text Random text is getting dropped from Wikipedia oneboxes. See . Screenshot:  This occurs regardless of HTTP/HTTPS and of whether a redirect URL is entered into chat. I initially thought the bug is triggered by &lt;span&gt;s or other HTML formatting elements, but in the Burj Khalifa article's HTML source code, there is no formatting around one of the deleted words. Possibly related to , but this is not specific to numbers. Glucagon stimulates glycogenolysis and gluconeogenesis, thus increasing the plasma glucose concentration — so that tissues get enough glucose in the fasting state. However glucagon also inhibits glycolysis. In this case, what is the point of having high blood glucose if it cannot be catabolized? Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set Problem 4.3.32 in Linear Algebra, Lay:  Let $V$ and $W$ be vector spaces, let $T:V\to W$ be a linear transformation, and let $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ be a subset of $V$. Suppose that $T$ is a one-to-one transformation [...]. Show that if the set of images $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set.  All I can think of doing: If $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then there exist weights $c_1, \dots, c_p$, not all zero, so that $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = \mathbf{0}$$ Since $T$ is linear: $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = T(c_1\mathbf{v}_1) + \cdots + T(c_p \mathbf{v}_p)=T(c_1 \mathbf{v}_1 +\cdots + c_p \mathbf{v}_p)=\mathbf{0}$$ So the set $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ must be linearly dependent. Am I missing something? And why must $T$ be on-to-one for this to be true? An example showing that it is not true if $T$ is not one-to-one would be great! I am getting my feet wet with Git and have the following issue:  My project source tree:  / | +--src/ +----refs/ +----... | +--vendor/ +----...   I have code (currently MEF) in my vendor branch that I will compile there and then move the references into /src/refs which is where the project picks them up from.  My issue is that I have my .gitignore set to ignore *.dll and *.pdb.  I can do a git add -f bar.dll to force the addition of the ignored file which is ok, the problem is I can not figure out to list what files exist that are ignored.  I want to list the ignored files to make sure that I don't forget to add them.  I have read the man page on git ls-files and can not make it work. It seems to me that git ls-files --exclude-standard -i should do what I want.  What am I missing? In PHP you can just use $_POST for POST and $_GET for GET (Query string) variables. What's the equivalent in Python? Shadow Incorrect at Bottom of Edge In the images below, I have a cube that is sticking up about two thirds through a plane. But, despite the fact that the cube is not hovering over the plane, the shadow seems to indicate otherwise. How can I fix this shadow? Copy the contents of a file into the clipboard without displaying its contents How to copy the contents of a file in UNIX without displaying the file contents. I don't want to cat or vi to see the contents.  I want to copy them to clipboard so that I can paste it back on my windows notepad.  I can't copy the file from that server to another due to access restrictions. Find the Unit Vector of a Three Dimensional Vector Is there a website that lets you shop for hotels with taxes and fees included in search results? Every time I book a hotel online, the price increases by 25% in "taxes and fees" at checkout, compared to the price featured in search results.  This does not occur when booking flights (I actually think there's a regulation against it for flights).  Is there a website that lets you compare the ACTUAL price of hotels with taxes and fees included, and also lets you book? Who in the DC Universe can stop Superman if he has a breakdown? Is there anything or anyone in the DC universe that can defeat Superman just in case he goes berserk on everybody?  Maybe Batman stores kryptonite in his Cave or something. I don't know if Batman actually has kyptonite, the example was just a theoretical example.  Who can stop Superman if needed, and how? Labeling linguistic examples with language information Where were the cannons in final fight in Wakanda in Infinity War? In Black Panther, Ulysses Klaue describes his arm cannon as a leaf blower in comparison to the cannons/weapons that Wakanda has.  Infinity War comes after the events of Black Panther, so the weapons should exist. During the final fight with the children of Thanos and Space Dogs (as Rocket refers to them), Wakandans could have used these powerful cannons, pointing them at the opened section of barrier. Instead they chose a fist fight.  Even M'Baku says that it would be the end of Wakanda, so the magnitude is huge and risks are very high. Why were these powerful weapons not used?    Edit 1: I watched the movie again and planes are there in the fight. But they were only used to destroy the spaceships that crashed. How do I restore .bat files association with the system (make them run when double-clicked)? What happens in electron-electron collisions? Why do birds sitting on electric wires not get shocked? When we touch electric wires, we get shocked. Why don't birds sitting on electric wires not get shocked? Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric?  $$R = \{(1, 2), (2, 3), (3, 4)\}$$  Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then $a = b$. In my textbook  it says the above is antisymmetric which isn't the case as whenever $(a,b)$ is in $R$, $(b,a)$ is not. Is there a way to add a timer to a Beamer presentation? Difference in meaning and use of future perfect forms So uh i've got this problem with a penetrating object and another object that serves as something like a condom if you will    So what I want, is for the penetrating object to not deform while moving into this "rubber cylinder", while the rubber cylinder does deform and reshape itself to the outer form of the penetrating object. That is, if the pen. object is bigger than the entrance hole ofc.  Also, when the object leaves the "rubber cylinder" I want the outer edges to follow (mid object in picture) and then jump back (right object in picture)    Parenting the size and position of the "rubber cylinder"-entrance to the position of the penetrating object would work i guess, but I think it'd create problems when the pen. object enters from another angle (not perpendicular to the entrance). I just read another post where someone mentions Vertex Weight Proximity modifier in a combination with a cast modifier. Don't know if thats gonna work yet. Couldn't find the time to test it. Is the money you get from shorting a stock free to use for going long on other stocks? I shorted a particular stock which resulted in $10,000 being deposited into my Interactive Brokers account. I then put all of that $10,000 into VTI (total stock market index). Basically making the bet that the stock I shorted will perform worse than the total stock market.   I assumed I would be charged some sort of interest or fees for using that money, however it appears that I am not being charged at all. Are there really no costs for doing this?