Evaluate  $$\binom{n}{1} + 2\binom{n}{2} + 3 \binom{n}{3}+...+ n \binom{n}{n}$$ My attempt : I know  $(a+b) ^n = \sum\limits_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$ But here Im confuse how to evaluate and im not able to proceed further.
I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality.
The closest I can get to making mesh lights affect their environment in Blender Eevee is giving them an Emission material and turning on "Bloom". But this isn't emitting proper light, it just looks like it is, but it doesn't light the surroundings whatsoever.     I would like to be able to light a scene with meshes without having to put point lamps everywhere just to make them emit light. It may be something that will happen to the person who is having the feeling but it may also be to several persons, as might occur with a highly destructive earthquake, for instance.   The word or phrase would be used in the following sentence:      “I’m no spiritualist but I have a feeling of ___. I hope it’s just a feeling."   EDIT - The phrase I'm looking for conveys a much stronger feeling than "I have a bad hunch". It is about something catastrophic which the person feels is "going to happen in a few minutes". Making Ctrl+C copy text in gnome-terminal? Is it possible for me to make Ctrl+C perform a copy command if there is highlighted text in the terminal? Otherwise, it should retain its normal behavior.  (If there is a terminal that can do this other than gnome-terminal, that's probably fine too.) Reprojecting raster from lat/lon to UTM in R? i have to turn it into a  UTM in order to make the buffer functional.  wets&lt;-readOGR(dsn=".",layer="shapefile") r.raster &lt;- raster() extent(r.raster) &lt;- extent(wets) res(r.raster) &lt;- 100   wets.r &lt;- rasterize(wet,r.raster) plot(wets.r) wetsbuf&lt;-buffer(wets.r,width=500)   During the buffer creation which is the last line of code, it gives this warning:  Warning message:   In couldBeLonLat(x) :   raster has a longitude/latitude CRS, but coordinates do not match that   here's the info    summary(wets.r)           layer  Min.        1  1st Qu.     1  Median      2  3rd Qu.     9  Max.       11  NA's    52629  summary(wets)    Object of class SpatialPolygonsDataFrame Coordinates:       min       max  x  683705  714088.8  y 4326266 4343768.0  Is projected: TRUE   proj4string :  [+proj=tmerc +lat_0=0 +lon_0=24 +k=0.9996 +x_0=500000 +y_0=0 +datum=GGRS87  +units=m +no_defs +ellps=GRS80 +towgs84=-199.87,74.79,246.62]  Data attributes:    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.      0.0     2.5     5.0     5.0     7.5    10.0         wets.r  class       : RasterLayer  dimensions  : 175, 304, 53200  (nrow, ncol, ncell) resolution  : 100, 100  (x, y) extent      : 683705, 714105, 4326268, 4343768  (xmin, xmax, ymin, ymax) coord. ref. : +proj=longlat +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0  data source : in memory names       : layer  values      : 1, 11  (min, max) attributes  :    ID FID  from:  1   0  to  : 11  10   I have to change the prjection in order to be possible to do the buffer. Showing two ring homomorphisms that agree on the integers must agree on the rationals I have two ring homomorphisms $f,g\colon \mathbb{Q}\to X$. I know that $f=g$ on the integers. How can I show that $f$ and $g$ agree on the rationals?   My attempt:  let $x,y \in \mathbb{Z}$, $y\neq 0$.  $$\begin{align*} f(x/y)&amp;=f(x)f(1/y) &amp;&amp;\text{because }f\text{ is a ring homomorphism}\\  &amp;=g(x)f(1/y) &amp;&amp;\text{because }f\text{ and }g\text{ agree on the integers} \end{align*}$$  I do not know how to rewrite $f(1/y)$ so that I can replace it with $g(1/y)$. Any hints will be much appreciated. Thanks. Does anyone know of a program or way to log all network activity to a specific program on one's computer (program and/or service)?  I know I can see this "live" in Resource Monitor, but I'm wondering if anyone knows of a program or way to track and log all network activity to/from a specific process and/or service.  I'm finding a lot of "live"-type monitors, like Windows Resource Monitor, but nothing that can log. How can I convert a string to boolean in JavaScript? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks 64 bit enum in C++? Is there a way to have a 64 bit enum in C++?  Whilst refactoring some code I came across bunch of #defines which would be better as an enum, but being greater than 32 bit causes the compiler to error.  For some reason I thought the following might work:  enum MY_ENUM : unsigned __int64   {       LARGE_VALUE = 0x1000000000000000,   }; While looking over various PHP libraries I've noticed that a lot of people choose to prefix some class methods with a single underscore, such as  public function _foo()   ...instead of...  public function foo()   I realize that ultimately this comes down to personal preference, but I was wondering if anyone had some insight into where this habit comes from.  My thought is that it's probably being carried over from PHP 4, before class methods could be marked as protected or private, as a way of implying "do not call this method from outside the class".  However, it also occurred to me that maybe it originates somewhere (a language) I'm not familiar with or that there may be good reasoning behind it that I would benefit from knowing.  Any thoughts, insights and/or opinions would be appreciated. How do I install the latest OpenOffice? Proof of a formula for the power set of a union of two sets I need to prove that:  $2^{A\cup B} =\left \{ X\cup Y \mid X\in 2^{A}, Y\in2^{B} \right \}$  Note that $2^{A} = P(A)$ (The power set of A).  While I can prove the Right - Left direction fairly easily, It seems very hard for me to prove the Left - Right direction. Can I get some help with that?  How do I prove that $t\in2^{A\cup B}$ makes $t\in \left \{ X\cup Y \mid X\in 2^{A}, Y\in2^{B} \right \}$? What is $P(X > 0 \mid X + Y > 0)$ given that $X, Y$ are i.i.d standard normal? What does this red icon on my panel mean? This red icon.... It showed up after I upgraded to Ubuntu 12.04LTS; I can't figure out how to ditch it. It seems to be a notification thing.  I am at a loss.  The icon is this one:      I clicked it and read this:     An error occurred, please run Package Manager from the right-click menu or apt-get in a terminal to see what is wrong. The error message was:  'Error: BrokenCount&gt;0'.        This usually means that your installed packages have unmet dependencies. Is it true that $\lfloor\sqrt{n}+\sqrt{n+1}+\sqrt{n+2} \rfloor=\lfloor\sqrt{9n+7}\rfloor$? I know it's true that  $$\lfloor\sqrt{n}+\sqrt{n+1} \rfloor=\lfloor\sqrt{4n+1}\rfloor,\forall n\in \mathbb N. \tag 1$$ Is it true that  $$\left[\sqrt{n}+\sqrt{n+1}+\sqrt{n+2} \right ]=\left[\sqrt{9n+7}\right], \tag 2$$ $$\left[\sqrt{n} +\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}\right ]=\left[\sqrt{16n+20}\right], \tag 3$$ $$\left[\sqrt{n} +\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\right ]=\left[\sqrt{25n+49}\right], \tag 4$$ for all $n\in \mathbb N$? I have checked $(2),(3),(4)$ through $10^6.$   (I think $(4)$ maybe has counter-examples, but I cannot find it.) There exist several implementations of the GBDT family of model such as:   GBM XGBoost LightGBM Catboost.    What are the mathematical differences between these different implementations?  Catboost seems to outperform the other implementations even by using only its default parameters according to , but it is still very slow.  My guess is that catboost doesn't use the dummified variables, so the weight given to each (categorical) variable is more balanced compared to the other implementations, so the high-cardinality variables don't have more weight than the others. It allows the weak categorical (with low cardinality) to enter to some trees, hence better performance. Other than that, I have no further explanation. I've ran out of chocolate bars and didn't realize I could enchant my giant spoon and knight's armor. Is there any way to get more reliably? Movie where a guy is sent to a sandy planet where time moves really fast but slow to the outside observer All I remember is the guy was sent to a planet to test something or he did something illegal, and time moved really fast on the planet but slow to the outside world. He had to fix something to leave the planet I think. His dad was also dying and he checked how many days he had left every day, but in the end his dad didn't actually die as time was fast. It took place on a sandy planet. Number of triangles formed by $m$ lines While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it