Stars and bars but with distinct objects The total number of ways which 5 balls of different colors can be distributed among 3 persons so that each person gets at least one ball is..?  My attempt: So, first we can choose any three balls to allocate to any person, so $\binom{5}{3}$ and, then distribute them then in 3! ways. And, so we have, $\binom{5}{3} \cdot 3! $. Now I have two remaining balls and this I can give randomly to any of the three people. For the first ball I have three choices and so do I for the second ball. Hence,  The net number of cases is $ \binom{5}{3} \cdot 3! \cdot 3^2$ Now, I know that like the thing can happen in reverse. For example distributing three then distributing two, I could have some overlapping cases where the items which are distributed in the distributing two case is actually distributed in the distributing three step. How do I account for the overlapping case? Ans: 150 Stars and bars attempt: We have five objects and introduce two objects as dividers, subtract cases where one guy gets nothing (3 x 6!) and add cases where two people get nothing and third guy everything (3) What I am looking for : A fix to this method such that it gives me the correct answer, in the other stack post linked to this one, it has ample amount of methods based on other ways to solve it but my question here is how to fix this method.
Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball?   My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ which is equal to $66$?
How can I connect the top loop to the down loop with faces Hey I want to connect the bottom loop with the top loop so that it forms a solid.  I have tried selecting multiple vertices and pressing F but that just creates a large face which I do not want. I have tried using grid fill bu that doesn't work either. The solution that seems to work is by selecting four vertices and pressing F and then selecting the next 4 and pressing F and on and on. That just takes too long. I found there are so much excellent answer to questions about how to product high quality typesetting output. But I wonder how to save ink with LaTeX. What we need to think about to manage cost. page layout or something else? Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$ Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural numbers.  I am thinking about binomial formula $$\sum_{k=0}^n\binom nk x^{n-k}y^k=(x+y)^n$$ but I'm not sure how to use it.  I find this problem tantalizing because it looks as if there should be some sort of intuitive way so that is why I share it here. I am looking for an answer like  if possible. Can you find one? I want to appropriate line segments that share the same geometry with a particular line width and labels based on the aggregated values of one field "GIS@BUS_5_Min".  My attribute table and polyline objects look as following, see image below   "BUSLINIE" means a bus line, "Overlap" means if the line has overlaps, i.e. it's geometry repeats.  Values of the field "GIS@BUS_5_Min" for these three objects has to be aggregated as one value and represented as a label sum("GIS@BUS_5_Min").  And the width of this line (selected in yellow) will be sum("GIS@BUS_5_Min")/100 = 19.10/100 = 0.1910  How can I achieve such output?  I assume it is possible with Layer Properties &gt; Width &gt; Data defined override &gt; Edit..., but I do not know how to check the same geometry and how to sum the values of certain features with CASE WHEN condition THEN result END conditional function?  Dissolve function does not work in my case because I do not have any key-fields.    References: Is there a way to disable the hotkeys? Is it possible to make an underfull tabulary spread out? "The news is good." Why? "Undefined reference to" template class constructor What is the formula for percent of frame filled at a specific distance and focal length? How do I enable Remote Desktop connection on Xubuntu 11.10? Can I use Minecraft PC textures and worlds in Pocket Edition? Now that I've discovered that I can host a , I'd like to know if is it possible to use the textures and maps of the PC version for the pocket version.  I'm planning to visit Peru and Machu Picchu. It would be really cool to entice them for the voyage by letting them play a  in their favourite video game. Unfortunately, this God of Deceit world looks really sophisticated. I don't believe the Pocket Edition would be able to play it. What does the 'static' keyword do in a class? Applying for a Schengen Visa without itinerary I am currently residing in Ireland with work authorization permit. In the next few months I am planning for a trip to few schengen countries. I have not yet decided my itinerary. Is there any Schengen country embassy which can process a tourist Schengen visa for 90 days? I am trying to encrypt an entire USB drive with Truecrypt 7.1a. But as long as there is a partition it will only encrypt that. This will make it appear like the USB is full instead of completely empty, which isnt what Im aiming for.   I've tried with DISKPART, USB Partition Tool, and the built-in storage manager. When I delete all partitions, Windows still assigns the drive a letter and assumes that it has a partition covering the whole space.  I know that it is possible as I have done it with another USB, but I just don't remember how I did it.  Ideas? Let There Be Light Story Identification I am looking to identify a story that I read around 2001 (it is probably much older).  Someone showed me the story on pages printed from the internet, and it was attributed to Isaac Asimov (although I cannot state that is the true author).    The story is told in glimpses at different points of time in the future.  In each glimpse, humans are asking a more and more "intelligent" computer how to prevent the universe from collapsing into a singularity in the future, and the computer states it needs more computational power and resources to solve this problem.  At some point, man becomes one with the computer, and the computer exists in hyperspace (or subspace?).    In the final glimpse, it is the end of time and the universe is collapsing into a singularity.  At the instant the universe reaches the singularity, the computer figures out how to prevent it and says "Let there be light" and the universe begins to expand again.  Is anyone familiar with this story? What pins on a nRF51822 do I connect to a STM32 programmer? if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! If I know the index node (inode) of a file, but I don't know its path (or any of its paths), is it possible to create a hard link to that inode directly?  I could find the file using sudo find / -inum 123546 and then create a hardlink, but that would be way too slow for my application.  N.B. I'm using an ext4 filesystem. Difficulty in creating macro '\newcommand' for the beginning and end of frames in Beamer I would like to create a Latex macro for my beamer presentations. I use beamer frequently, and want to make macros for the \begin{frame} and \end{frame} to require fewer keystrokes. At the beginning of the document I have:  \newcommand{\bfrm}{\begin{frame}}%start new frame \newcommand{\efrm}{\end{frame}}%end new frame   Which compiles fine. I then test the first of the two macros for the beginning of the frame:  \bfrm{New frame testing}     blah blah \end{frame}   Which produces the result I am looking for. I then test the second of the macros:  \bfrm{New frame testing}     blah blah \efrm   Fails to compile, saying that there is a runaway argument. What is the reason for the failure? Using customized Coordinate System in ArcGIS Desktop? I don't know so much about coordinate systems...  In my office we use to deal with spatial data coming from archaeological sites. Each site has its own x-y-z coordinate system (GCS). Three simple ortogonal cartesian axis. In the last years we have been managing this spatial data through GIS software (ArcGIS), without using specific coordinate system (just leave it as "undefined")  I'd like to know if there exisits any GCS designed to deal with such datasets using simple cartesian orthogonal axis, without grid distortions of the typical GCS. In addition, I'd like to know if this system is suitable for using it in an online mapping application.  By the way, we manage 2D (ArcMap) and 3D (ArcScene) environments and work with "mm" as length base unit.   If such a thing doesn't exists, maybe someone knows how to create it.