How would you solve a problem: we have $30$ students in a classroom. How could we divide them into three groups, if there needs to be $5$ students in the first group, $10$ students in the second group and 15 students in the third group. The result is: $\frac{30!}{5!×10!×15!}$ . Why isnt it $\frac{30!}{3!×5!×10!×15!}$ since we have three groups with $3!$ permutations.
How many ways can you divide $9$ students into three unlabeled teams where one team contains $4$ people, one contains $3$ people and the last contains $2$ people? Unlabeled, meaning that groups with abc = bca = cba, etc  I understand how to do this if the teams are labeled:  $$\frac{9!}{4!3!2!}$$  But there is a term missing in the denominator when the teams are unlabeled and I'm having difficulty understanding how to calculate how many ways the teams can be organized.  There are $3!$ ways to organize the same first group of $3$, $4!$ ways to organize the same second group of $4$ and $2!$ ways to organize the last group of $2$. Why wouldn't you multiply $3!4!2!$ in the denominator?   For instance:  (ABC, DEFG) = (ABC, DEGF)  = (ABC, DFEG) = (ACB, DFGE), etc
I'm currently in the process of making a custom map and I want to add blocks that can only be placed on a certain colour clay.  This is the command I tried using.  /give @p cobblestone 1 0 {CanPlaceOn:["hardened_stained_clay"],Damage:3}   However, even with this command the cobblestone can be placed on any colour of stained clay. What command can I use for placing only on a certain block? How to run a command that requires sudo at login A is similar to $A^k$, then each eigenvalue of $A$ is a root of unity Let $A \in \mathbb{C}(n,n)$ and $k \geq 2$ be an integer such that $$A \sim A^k$$. Show that if $A$ is non-singular then each eigenvalue of $A$ is a root of unity.   Attempt: Since $A \sim A^k$, $$PA = A^kP$$ where $P$ is an invertible matrix. Since $A$ is invertible, $0$ cannot be an eigenvalue of $A$. Suppose $$Av = \lambda v \quad v \neq 0$$ then $$PAv = \lambda Pv$$ $$\therefore A^k(Pv) = \lambda (Pv) $$ but this implies that $Pv$ is an eigenvector of $A$. But the eigenvalues of $A^k$ are $\lambda^k$ $$\therefore \lambda^k=\lambda$$ which gives the conclusion required.   My questions is: Is the logic correct? If so, Am I missing any details? If not, then how could I approach this?   Thanks! Proving that closed (and open) balls are convex How do I read image files directly from a zip without extracting to disk? Is there an ideal PWM frequency for DC brush motors? I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands! Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$.  So from the given we know that there's an injection $f:B\to C$ and some functions $h:B\to A, g: C\to A$. We want to prove there's an injection $l_1:A\to C$. It appears that $f$ doesn't help here.  Trying to take representatives from $A$ and show they're in $C$ and there's an injection doesn't work so maybe the function should be $l_2: h \to g$ but I don't know how to work with it. I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function.  Well, I know that $P(T_i&gt;t)=e^{-\lambda t}$ and therefore $f_{T_i}(t)=\lambda e^{-\lambda t}$ so I need to find $P(T_1+\cdots+T_n&gt;t)$ and take the derivative. But I cannot expand the probability term, you have any ideas? Is it possible to do a cross page POST to an ASP.NET, from a static HTML form? Maximum Likelihood Estimation (MLE) in layman terms Could anyone explain to me in detail about maximum likelihood estimation (MLE) in layman's terms? I would like to know the underlying concept before going into mathematical derivation or equation. In recent versions of PostgreSQL (as of Dec 2013), can we share a query between two or more cores to get a performance boost? Or should we get faster cores? This next year, I am helping an effort to clean several SQL Server environments.    We have about 10,000 stored procedures and estimate that only about 1000 of them are used on a regular basis, and another 200 or so are used on a rare occasion, meaning we have a lot of work to do.    Since we have multiple departments and teams that can access these databases and procedures, we are not always the ones calling the procedures, meaning that we must determine what procedures are being called.  On top of that, we want to determine this over a few months, not in a few days (which eliminates some possibilities).  One approach to this is to use the SQL Server Profiler and track what procedures are being called and compare them to the list of what procedures we have, while marking whether the procedures are used or not.  From then, we could move the procedures to a different schema in case a department comes screaming.  Is using the Profiler the most effective approach here?  And/Or have any of you done something similar and found another way/better way to do this? And now I am unable to chmod it back.. or use any of my other system programs. Luckily this is on a VM I've been toying with, but is there any way to resolve this? The system is Ubuntu Server 12.10.   I have attempted to restart into recovery mode, unfortunately now I am unable to boot into the system at all due to permissions not granting some programs after init-bottom availability to run- the system just hangs. This is what I see:  Begin: Running /scripts/init-bottom ... done [   37.062059] init: Failed to spawn friendly-recovery pre-start process: unable to execute: Permission denied [   37.084744]  init: Failed to spawn friendly-recovery post-stop process: unable to execute: Permission denied [   37.101333] init: plymouth main process (220) killed by ABRT signal   After this the computer hangs. Is there any hint that the MoM or anywhere else got support in combating voldemort the first time around? Or is everything completely UK-centric?  I can't remember anyone in the second war being specifically sent from europe to help, even though we see international wizards in book 4. But I don't think all the events of the first war are covered in the books.  So are other bodies like the MoM involved in the first war, or was it isolated in the UK? At what stage should acknowledgements be inserted into the paper if review is not double-blind? When the review process is double-blind, I guess (although I have not read this anywhere) that acknowledgements should not be written in the submitted paper, as this might break anonimity. They should only be inserted into the camera-ready version, after the paper is accepted.  What is the custom when the review process is not double-blind? Should I write the acknowledgements in the submitted version, or wait to the camera-ready version? I have a website the seems to load very slowly. When I run a speed test on it I see there seems to be a 6 secound gap before the HTML loads. Images and JS scripts load very fast after that point.  You can see in the pic below the yellow 'wait time' bars:    This seems to be consistant no matter what the HTML content on the page is.  This site is using a CMS (ModX Revo) so HTML is actually stored in an SQL database and brought forth by PHP, but I've never experienced this issue before.  Would anyone know what is causing this and how I could speed it up? Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data I know this is a fairly specific R question, but I may be thinking about proportion variance explained, $R^2$, incorrectly. Here goes.  I'm trying to use the R package randomForest. I have some training data and testing data. When I fit a random forest model, the randomForest function allows you to input new testing data to test. It then tells you the percentage of variance explained in this new data. When I look at this, I get one number.  When I use the predict() function to predict the outcome value of the testing data based on the model fit from the training data, and I take the squared correlation coefficient between these values and the actual outcome values for the testing data, I get a different number. These values don't match up.   Here's some R code to demonstrate the problem.  # use the built in iris data data(iris)  #load the randomForest library library(randomForest)  # split the data into training and testing sets index &lt;- 1:nrow(iris) trainindex &lt;- sample(index, trunc(length(index)/2)) trainset &lt;- iris[trainindex, ] testset &lt;- iris[-trainindex, ]  # fit a model to the training set (column 1, Sepal.Length, will be the outcome) set.seed(42) model &lt;- randomForest(x=trainset[ ,-1],y=trainset[ ,1])  # predict values for the testing set (the first column is the outcome, leave it out) predicted &lt;- predict(model, testset[ ,-1])  # what's the squared correlation coefficient between predicted and actual values? cor(predicted, testset[, 1])^2  # now, refit the model using built-in x.test and y.test set.seed(42) randomForest(x=trainset[ ,-1], y=trainset[ ,1], xtest=testset[ ,-1], ytest=testset[ ,1]) What is the music on S3E16 when Todoroki was fighting group of ninjas? Is the universe really "fine-tuned for life"? I understand what is meant when we say , but I've seen it claimed it's is fine tuned "for life". Is it?  When I look at known or even speculated life in our universe, I see life clinging to a tiny crumbs separated by vast voids. Our life needs baryonic matter, only 5% of the universe. Its density is $10^{-30}  g/cm^3$, but it clumps together leaving vast voids between the clumps. 99% is in stars and black holes and nebula utterly inhospitable to known life.  That leaves the parts "fine tuned for life" to be 0.05% of the universe. The planets, asteroids, and comets seem like just debris separated by  than the life-sustaining clumps preventing life from spreading.  Even that seems to be used very inefficiently. Life requires the energy of a star, but it's spewed out in all directions. We only receive a tiny fraction of it. Similarly, life requires heavier elements created in supernova, but these are also spewed out in all directions. And it took billions of years to develop a being capable of asking "is the universe fine tuned for life?"  It seems like we live on the crumbs of creation warmed by a rounding error of the heat from our star. Taking this view, the universe appears extremely inhospitable to life. It could be a lot better tuned.  Is it that the universe is fine-tuned for life? Or is it that the universe is so large and life requires such an infinitesimal fraction of the universe that it's a near guarantee were we to roll the dice again that some crumb, maybe not even baryonic matter, could develop something complex enough to ask "why is the universe fine tuned for life"? Am I just restating the weak anthropic principle?  I understand this is straying into philosophy. If that's off topic, are my  physical assumptions correct?