In how many ways can eight distinct balls be distributed to three distinct boxes so that each box must contain at least one ball?   So in order to approach this problem I was first thinking of assigning one ball to each box. Since there are 8 total balls I have 8C3 ways of picking 3 of them. Out of those 3 that I picked they can be arranged in 3 different ways since there are 3 boxes. Now I have to arrange the remaining 5 balls. This can be done in 5x5x5 ways. Then I multipled 3(8C3)(5x5x5) to get my final solution. However my textbook says the solution should be 5796. What is wrong with my approach/what adjustments can be made to make my approach work?
Find the number of ways in distributing $8$ distinct balls into $6$ distinct boxes such that there is at least $1$ ball in each box.  We are well acquainted with the traditional Inclusion-Exclusion principle in solving this and it will come up as $191520$ ways.  I tried in this way:  When we distribute $8$ balls into $6$ boxes we have two possible cases  Case $1.$ There are exactly $2$ boxes with $2$ balls each  Case $2.$ There is exactly $1$ Box with $3$ balls in it.  For Case $1$, the distribution is done by choosing $2$ boxes from $6$ boxes and $2$ balls from $8$ balls and those two balls can be arranged in $2!$ ways. But I require another two balls from remaining $6$ balls to be placed in each of these two boxes which can be done in $\binom{6}{2} \times 2!$ ways. Finally remaining $4$ balls can be distributed in remaining $4$ boxes in $4!$ ways. So total number of ways for Case $1.$ is  $$\binom{6}{2} \times \binom{8}{2} \times 2! \times \binom{6}{2} \times 2! \times 4!=604800$$  For Case $2$, the number of ways is  $$\binom{6}{1} \times \binom{8}{3} \times 5!=40320$$  I am pretty sure the value obtained for Case $2.$ is correct. but I could not find where I went wrong in Case $1.$
In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?  I tried to draw diagrams onto a $8\times8$ square but I'm only getting $16$ ways.  Does that sound right?  Thanks for the help! Explicitly accept change for PPA 'Label' When I run apt-get update on my ubuntu 17.10 I receive the following error:  E: Repository 'http://ppa.launchpad.net/ondrej/php/ubuntu artful InRelease' changed its 'Label' value from '***** The main PPA for PHP (5.6, 7.0, 7.1) with many PECL extensions *****' to '***** The main PPA for supported PHP versions with many PECL extensions *****' N: This must be accepted explicitly before updates for this repository can be applied. See apt-secure(8) manpage for details."   The man page doesn't really point to a solution.  My question is, how do I explicitly accept the change? A $60\times 60$ board is partitioned into rectangles of size $2\times 5$ (or $5\times 2$). Is it true that there always exist another partition into rectangles of size $1\times 3$ (or $3\times 1$) such that any $2\times 5$ (or $5\times 2$) rectangle contains a $1\times 3$ (or $3\times 1)$ rectangle?  For the "simple" partition into $2\times 5$ rectangles, this is certainly true: use the simple partition into $1\times 3$ rectangle.  Another way to partition is to first partition the $60\times 60$ board into $12\times 10$ boards, and then for each such board, put two $2\times 5$ at the top and ten $5\times 2$ below. It is not hard to tile each such $12\times 10$ board with $1\times 3$ (or $3\times 1)$ rectangles so that each $2\times 5$ (or $5\times 2$) contains a $1\times 3$ (or $3\times 1)$ rectangle.  (I'm assuming that an $m\times n$ rectangle has $m$ rows and $n$ columns.) This is probably easier to show than tell...   On the recent question , I see the following:    As you can see, the first five links render properly, but after that, they stop converting.  Unfortunately, this isn't obvious to the person writing the post, which I discovered when I went to edit the post to fix the "broken" links. In the edit window, it renders all of the links just fine:    I understand that this behavior is expected based on balpha's answer :        Links to questions on the same site are unlimited.   Links to other sites are limited to             a) 5 linked-to sites and     b) N questions per target site, where N is 200 on meta sites and 10 on non-metas (this is about the site that the post containing the link is posted on, not the site that is being linked to)         So if this rule is still going to stay in effect, please fix the edit view to also not automatically convert more than five linked-to sites to help people writing posts know that they need to manually title the links to the other questions.  That said, I'd prefer that (at least on MSE) this limitation be removed or raised. Even in the answer, balpha says:     The restrictions are currently as follows, but note that they do look a bit contrived, resulting from the fact that they have grown over time (from single site to cross-site, from "there are no subsites" to "facebook.SO", etc.), so it's not unlikely that we'll revisit them.   So, if it's not hurting performance, can we please increase the limit or remove it? Origin of "Fits [x] to a T"? The above phrase is something I've known for as long as I can remember, though I don't know from where. What is its origin and usage? Example: &quot;While Android products aren't quite what I need, iOS devices fit my needs to a T.&quot; SQL Server 2000 on Windows 10 I need to install SQL Server MSDE 2000 SP4 on Windows 10.  I have other machines that run Windows 7 and 8.1 and SQL Server MSDE 2000 works fine (with merge replication configured too).  Now I testing the same scenario with Windows 10, but setup is closing unexpectedly during installation.  I know that this version is very old, but is there a workaround to install it on Windows 10? How do exchanges match limit orders? Is Duck/Duct tape safe to use to insulate mains wires? Question says it all really. Based in the UK so mains is 240v and 20 amps. It's a lighting circuit. "...his parents' dream of *him* achieving a Cambridge degree." What is the function of "him" here? As the question title suggests, how do I see that every left ideal of a square matrix ring over a field is principal? Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? Boot failure : No DEFAULT or UI configuration directive found! How to set Windows bootloader as default bootloader? I have a Win 7/11.10 dual boot system with Grub 2 as the default bootloader. However, I want to make Windows bootloader the default bootloader (grub 'dos' like interface scrares the hell out of my sister) so when I start the PC I will have the following boot options: 1. Windows 7 2. Ubuntu  NB: A process to reverse this operation so Grub can be the default bootloader again will also be appreciated Category-theoretic limit related to topological limit? Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...?  To be more precise, is there a category-theoretic setting of some non-trivial topological space such that these different concepts of term 'limit' somehow relate?  This question came to me after I saw (  ) a surprising fact that generalised metric spaces can be seen as categories enriched over preorder $([0,\infty],\leq)$. I found the following question about compactness:     A set $S\subseteq \mathbb{R}^n$ has the following property:   if a sequence of $S$ has exactly one accumulation point in $S$, then it converges in $S$. Prove that $S$ is compact.    We know that in $\mathbb{R}^n$ compact sets correspond to closed and bounded sets.   (1) I am trying to prove that $S$ is closed. The condition in the question says if a sequence of $S$ has an accumulation point in $S$ then it converges in $S$. I think instead the condition should be if a sequence of $S$ has an accumulation point( in $\mathbb{R}^n$, not necessarily in $S$ )then it converges in $S$.  Am I correct?  (2) How to prove that $S$ is bounded?  P.S. I have made an edit (there is exactly one accumulation point).  Source of the question:  of this problem set. Tabular with p type columns to fill page width Poisson Distribution of sum of two random independent variables $X$, $Y$ $X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to derive it. How do I change my timezone to UTC/GMT? Currently when I execute the date command from bash I get the following displayed  Thu May 17 12:21:00 IST 2012  Below is my timezone config based on executing dpkg-reconfigure tzdata     Current default time zone: 'Europe/Dublin'   Local time is now: Thu May 17 12:32:52 IST 2012.   Universal Time is now: Thu May 17 11:32:52 UTC 2012.     Any idea how I change from IST to UTC/GMT? Does anyone have a Terminal command-line recipe for making a new bootable USB flash drive for one of the new MacBook Air's?  Is there a way to dd copy the USB boot device that comes in the box and make the new copy bootable?  I'd like to make a new backup copy of the USB rescue device for my MBA, preferably on a much larger flash drive with room for a backup of some of my working directories.  Added:  And is it possible to make, not just a rescue drive, but a bootable USB drive that will boot a fully updated version of the OS (say 10.6.8 or later)?  If so how on a MacBook Air 11, and how big a USB stick will this require? How to count the number of set bits in a 32-bit integer? 8 bits representing the number 7 look like this:  00000111   Three bits are set.     What are algorithms to determine the number of set bits in a 32-bit integer?