Combinatorics: Picking n marbles from 3 different groups of marbles.
Number of ways to choose $n$ balls from $3n$ balls, distinguishable/indistinguishable.
Vandermonde Determinant This is an exercise from Ian Stewart's Galois Theory, $3^{rd}$ edition:     If $z_1,z_2,\ldots,z_n$ are distinct complex numbers, show that the determinant $$D=\left|\begin{array}[cccc]  11&amp;1&amp;\cdots&amp;1\\ z_1&amp;z_2&amp;\cdots&amp;z_n\\ z_1^2&amp;z_2^2&amp;\cdots&amp;z_n^2\\ \vdots&amp;\vdots&amp;\ddots&amp;\vdots\\ z_1^{n-1}&amp;z_2^{n-1}&amp;\cdots&amp;z_n^{n-1} \end{array}\right|$$   is nonzero.       Hint: Consider the $z_j$ as independent indeterminates over $\Bbb C$. Then $D$ is a polynomial in the $z_j$, of total degree $0+1+\cdots+(n-1)=\frac{1}{2}n(n-1)$. Moreover, $D$ vanishes whenever $z_j=z_k$ for $k\neq j$, as it then has two identical rows. Therefore $D$ is divisible by $z_j-z_k$ for all $j\neq k$, hence it is divisible by $\prod_{j&lt;k}(z_j-z_k)$. Now compare degrees.   My question is how to do this. I follow the hint and I'm thinking of $D$ as a cofactor expansion; is that the idea? Also, the fact that $D$ vanishes whenever $z_j=z_k$ for $k\neq j$ follows since it would have two identical columns, right? I don't see why $z_j=z_k$ would imply it has two identical rows, as the hint suggests. The fact that it is divisible by $(z_j-z_k)$ is easy since that would make $z_k$ a root for $k=j$, but I'm not quite sure I follow why the degrees would be different. I think the degree of the product would be $1+2+\cdots+(k-1)$, but then add these up for all $k&lt;n$?   I appreciate any clarification that you can provide! Colors in Photoshop CS6/CC look different than in browser? One of my friends, a graphic designer told me recently that sometimes colors in Photshop look slightly different than the ones rendered in browser.  I know it is possible when you save a file and then use it on a website due to palettes, but he uses given color in PS, for example #b81e1e, then he creates a DOM element and sets its background to #b81e1e and the color is rendered differently in browser than in PS (I guess it's not that subtle as he can easily spot it). It's not a regular thing and it doesn't happen always, but he states it happened to him a few times, although he can't really tell me with witch colors.    Is it even possible? Because I have never really heard about anything like this &amp; nothing like this ever happened to me... Non-linearity and self-coupling of gravity I have heard that non-linearity of  has to do with the fact that gravity self-couples. What does non-linearity have to do with self-coupling? From 8th June there's been a  period for people entering the UK. From 10th July, travellers from  via a travel-corridor. If more countries are added to the list of exemptions then the date for each country will be later than the 10th July. If a traveller entered the UK from a country that is not on the exempt list and that country is added to the exempt list prior to the end of the 14 day self-isolation period does the self-isolation period end early? For example, if an individual entered the UK from France on the 5th July, should they remain in isolation for 14 days or does the isolation end on the 10th when France is added to the list of exempt countries? How to measure time taken by a function to execute I need to get execution time in milliseconds.  I originally asked this question back in 2008. The accepted answer then was to use  However, we can all agree now  that using the standard  API is more appropriate. I am therefore changing the accepted answer to this one. Here's my attempt at it:  $query = $database-&gt;prepare('SELECT * FROM table WHERE column LIKE "?%"');  $query-&gt;execute(array('value'));  while ($results = $query-&gt;fetch())  {     echo $results['column']; } Changes to parameter not showing on Report Server after deployment How to make pdfsam work in (L)ubuntu 18.04? Rendering of Favorite/Ignored Tags in Firefox I have spotted a bug in rendering of Favorite Tags and Ignored Tags section in Firefox. This rendering issue is present on both stackoverflow.com and meta.stackoverflow.com sites.  I'm using Firefox 17.0.1 on Windows 7 x64. In Google Chrome page is rendered correctly.  Screenshots:  Firefox (please note cropped right part of both Add buttons)    Chrome (buttons are rendered correctly) How to count number of bases and subspaces of a given dimension in a vector space over a finite field? Let $V_{n}(F)$ be a vector space over the field $F=\mathbb Z_{p}$ with $\dim V_{n} = n$, i.e., the cardinality of $V_{n}(\mathbb Z_{p}) = p^{n}$. What is a general criterion to find the number of bases in such a vector space? For example, find the number of bases in $ V_{2}(\mathbb Z_{3})$. Further, how can we find the number of subspaces of dimension, say, $r$?   I need a justification with proof. I have a formula, but I am unable to understand the basic idea behind that formula. Recently, someone analyzed  and found that over half of the professors at the top 51 universities graduated from a top 10 university.  have also brought this up. From my personal observations, most schools do hire graduates from better ranked schools.  Is this because of the competitive job market? We have so many good applicants, we have to narrow it down some how! Or is it simply that these schools produce the most PhDs? Has this always been the case? How rare are exceptions to this? I know of a few people who graduated from a top 75 school and got hired at a top 50 school. But what about bigger gaps? The top 10 schools seem to just swap graduates, do they ever hire from a 50+ ranked school?  Update 2018: I have accepted a tenure-track position at a top 75 department at an R1 university immediately after graduating from an unranked department at an R2 university. It does happen! This may or may not generalize to other fields and countries. Why am I not getting a java.util.ConcurrentModificationException in this example? "Questions that may already have your answer" box overlaps onto the footer I’ve run into a limitation in the cURL bindings for PHP. It appears there is no easy way to send the same multiple values for the same key for postfields. Most of the workarounds I have come across for this have involved creating the URL encoded post fields by hand tag=foo&amp;tag=bar&amp;tag=baz) instead of using the associative array version of CURLOPT_POSTFIELDS.  It seems like a pretty common thing to need to support so I feel like I must have missed something. Is this really the only way to handle multiple values for the same key?  While this workaround might be considered workable (if not really annoying), my main problem is that I need to be able to do multiple values for the same key and also support file upload. As far as I can tell, file upload more or less requires to use the associate arravy version of CURLOPT_POSTFIELDS. So I feel like I am stuck.  I have  about this problem in more detail on the cURL PHP mailing list in the hopes that someone there has some ideas about this.  Suggestions or hints on where I can look for more information on this are greatly appreciated! Getting different answers when integrating using different techniques Is there a canonical way to typeset such quantities in equations and inline math?  \documentclass{article} \begin{document} $Re$, $Nu$ are clearly not an option. $\mathit{Re}$, $\mathit{Nu}$ appear ok to me. \end{document} Why I cannot ping 192.168.0.108 (my computer IP?) Finding languages such that $L_{1} \subseteq L_{2} \subseteq L_{3}$ where $L_{1}, L_{3} \notin \mathbb{R}$, $L_{2} \in \mathbb{R}$ I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that  $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin  \mathbb{R}$ and $L_{2} \in  \mathbb{R}$.  I know they exist, I need help finding them. a) There's a radially symmetric charge density $\rho(r)$ centered around the origin. Determine the dipolemoment of that charge density. b) Let $\rho(r)$ be an arbitrary charge density now. Under what circumstances does the dipole moment of the displaced charge density  $\rho '(\vec{r}) = \rho (\vec{r}-\vec{b})$ differ from the one not displaced at all.  Here were my ideas so far: a) Just thinking about the situation it has to be zero, right? I mean, since there's no real dipole. But how do I show that mathematically? I was thinking of just going like this (it may be wrong): Let the charge density be $\rho (r)=kr$, then we can get the charge q by integrating: $$q=4\pi \int_0^R kr\cdot r^2dr=\pi k R^4$$ I'm looking at the charge distribution as a spherical electron cloud with radius $R$. Then, since $p=qd$ and $d$ is zero because there are no two different charges the dipole moment is zero. Is that sufficient as an answer? b) I don't know how to approach this one. My guess is that its dipole moment is also zero because we're only looking at a displacement here. Anyone got any idea? I would appreciate any advice on this. I want to make an animation in time to a song so when I hit play on the timeline I need to be able to hear the song so I can finely tune the animations to start and end at the right time.  Is it possible to have sound play on an animation while it is playing in the 3d viewport before it has been rendered.