In reading about , I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to prove it. $$\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!.$$ After some reductions (including $\binom{k+1}{j}=\binom{k}{j}+\binom{k}{j-1}$) and "simplifications", I am still stuck and have no clues how to proceed. I do not know if this is true, either, though I think it should be.    Any help or reference is sincerely welcomed, thanks in advance.
The successive difference of powers of integers leads to factorial of that power. Here's the formula:     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$   Can anyone give a proof of this result?  Note: The original question was to prove the more general     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$$   for an integer $l$, which some of the answers address.
Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. When working with an image in Photoshop, I'd like to have a simple keyboard shortcut, like in , that automatically does:   Export > Export As or Export > Save For Web  Resize to biggest dimension = 600px (either width or height) Add suffix _600.jpg to filename   How to do that?  I'm not sure how to edit a recorded macro to do that (add the suffix and choose 600px for the biggest dimension).  Example: you open myphoto.png, you press F6 and then you automatically get myphoto_600.jpg. Is there a route that would allow travel to the US from the UK or Schengen area, given the US travel ban? I'm a European citizen living in the US on a work visa. Due to some unfortunate circumstances, I need to urgently visit my home country. However I would like to get back to the US after my visit is complete. Flying from the US to Europe is not a problem as there are still daily flights between some city pairs. However going back could be tricky, as  anyone who has been in the Schengen area in the past 14 days from re-entering the country.   Is there a flight route that would let me from Europe to some country, stay there for 14 days and then fly to the US? Canada is out, as they've banned international visitors. Mexico seems to keep their border open, but I  any direct European flights from their airports. Almost every other country either suspended flights or doesn't allow foreign nationals to enter. Problem with typesetting maths content in an unusual font in beamer Is there a way to block access to or wipe my Google data for a lost/stolen phone Other word for person who does something to others but does'nt like it done to them This is a duplicate question but has still not been answered for me. I'm so close to the exact word - I know there is one out there - but the closest word I can find is a hypocrite --     a person who claims to have certain moral principles or beliefs but behaves in a way that shows they are not sincere.    The word I'm looking for is a person who does not claim things, but does thing to another and they consider it wrong or immoral to have done to them. It seems common that PBX and other telephone hardware use a positive-ground power supply, where the "hot" line is at -48v.  What's the reason for that? Convert python datetime to epoch with strftime Usage of -n and -z in test built in - Bash surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism?  I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to preserve the property of being immersion.  I also noticed that if we replace $\mathbb{R}^2$ with $S^2$ or any other compact simply connected manifold, no such immersion exists, because then it would be a covering map (as a local diffeomorphism from compact space, thus proper), but since the manifold is simply connected, it must be a diffeomorphism. I just installed Call of Duty 4 and when I try to run the single player mode nothing happens. I just see a quick flicker on the desktop but the game doesn't launch. When I try to run it again, it gives me this message:      It appears that cod4 did not quit properly the last time it ran. Do you want to run the game in safe mode?    Multiplayer seems to work fine and when I try to switch to single player from the multiplayer menu the same thing happens, it just goes back to my desktop and that message shows up again Is it possible to use "negative sound waves" to "cancel out" a sound to create silence? Prove normal matrix is 4th power of some self-adjoint matrix Let $A\in\mathbb{M}_n(\mathbb{C})$ be a matrix of order $n\geq 2$. Let $p_A(x) = (x-\lambda_1)\cdots (x-\lambda_n)$ be the characteristic polynomial of $A$ such that all $\lambda_i$ are positive real numbers.  Suppose that $A$ is a normal matrix. Prove that one can write $A=G^4$ for some self-adjoint matrix $G$.  My thinking is that since $A$ is normal, by Principal Axis Theorem, we have $A=UDU^\ast$, where $U$ is a unitary matrix and $D$ a diagonal matrix. However, I fail to see how to proceed. Could anyone help me out? Early 80s anthology book about human kind being feared by other races in space I always thought the book was Manifest Destiny but its not. The story that sticks in my head was a human in a cage over a pit? Aliens walk in and discuss how dangerous humans were. He eventually escapes. Man I miss this book. wish I wouldve written down every book I ever read. Ugh Can I start and stop a tile floor job? I am tiling my bathroom floor and don't soon have a full day to dedicate to the job. Is it ok to lay down a few tiles each day and come back to the job when I have time? I would only lay mortar for the tiles I have time to lay each day.    My concern is that the mortar wouldn't properly "join" to the previously laid sections of tile and cause an inferior result, but I'm not really basing this concern on any facts. That is what I'm hoping to learn with this question. Why is the last digit of $n^5$ equal to the last digit of $n$? I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes.  Proposition: any permutation in $S_n$ can be written as a product of disjoint cycles.  Proof (attempt): $s\in S_n$. Let $\langle s\rangle$ act on $X=\{1,\cdots,n\}$. Define $x\sim x'\iff x'\in\text{Orb}(x)$ for $x,x'\in X:$ this is clearly an equivalent relation, hence the orbits partition $X$. Each orbit corresponds a cycle in $s$, so we are done.  Does this hold, or have I missed something vital? Books for starting with analysis I am interested in self-studying real analysis, and I was wondering which textbook I should pick up. I know all high school mathematics, I have read  (I did most of the exercises).  I have completed a computational calculus course which covered everything up to and including integration by parts (including the substitution method and Riemann sums) I am currently considering:    From what I have heard this is not very well suited for self-study and that while the exercises are extremely difficult, if you take the time they are worth the effort.    I have heard that while Spivak explains proofs in much more detail than Principles, it doesn't cover all of the material in the latter.    I don't know much about this. I have only seen some comments saying that it is an excellent introduction to analysis.    Extra clarification edit: I would prefer a book not to ''dumb down'' the material, something that would not hold my hand through every step, something that would force me to fill in the gaps myself instead of explaining every single step. That is why I am currently leaning towards Rudin, but before I decide I would still like some information on the book by Apostol. I have a register \count0=&lt;some-value&gt; and need to get the letter of the alphabet on the index of \count0. That is, if it is 1, I need an A, if 2, a B, etc.  Is this possible?  I'm reluctant sharing an MWE as it seems to overcomplicate things a bit:  \begin{tikzpicture} \node[regular polygon, regular polygon sides=7, minimum size=5cm] (A) {}; \foreach \i in {1,...,7}{     \count0=8     \advance\count0 by -\i     \node[circle, label=above:\scriptsize\the\count0, fill=black, minimum size=4pt, inner sep=0pt] (\the\count0) at (A.corner \i) {}; } \end{tikzpicture}   With the tikz package and the shapes.geometric tikz library.  This draws seven nodes labelled 1 .. 7 on a circle. I need to have A .. G instead of 1 .. 7. I see questions come up quite often that have this underlying issue, but they're all caught up in the particulars of a given feature or tool. Here's an attempt to create a canonical answer we can refer users to when this comes up - with lots of animated examples! :)    Let's say we're making a first-person camera. The basic idea is it should yaw to look left &amp; right, and pitch to look up &amp; down. So we write a bit of code like this (using Unity as an example):  void Update() {     float speed = lookSpeed * Time.deltaTime;      // Yaw around the y axis using the player's horizontal input.             transform.Rotate(0f, Input.GetAxis("Horizontal") * speed, 0f);      // Pitch around the x axis using the player's vertical input.     transform.Rotate(-Input.GetAxis("Vertical") * speed,  0f, 0f); }   or maybe  // Construct a quaternion or a matrix representing incremental camera rotation. Quaternion rotation = Quaternion.Euler(                         -Input.GetAxis("Vertical") * speed,                          Input.GetAxis("Horizontal") * speed,                          0);  // Fold this change into the camera's current rotation. transform.rotation *= rotation;   And it mostly works, but over time the view starts to get crooked. The camera seems to be turning on its roll axis (z) even though we only told it to rotate on the x and y!    This can also happen if we're trying to manipulate an object in front of the camera - say it's a globe we want to turn to look around:    The same problem - after a while the North pole starts to wander away to the left or right. We're giving input on two axes but we're getting this confusing rotation on a third. And it happens whether we apply all our rotations around the object's local axes or the world's global axes.  In many engines you'll also see this in the inspector - rotate the object in the world, and suddenly numbers change on an axis we didn't even touch!    So, is this an engine bug? How do we tell the program we don't want it adding extra rotation?  Does it have something to do with Euler angles? Should I use Quaternions or Rotation Matrices or Basis Vectors instead?