If i  want to distribute n chocolates to m friends considering each friend can  get 0 to n chocolates(unfair distribution is possible), how many ways can i do it?
I came across this formula in a list:  The number of ways of distributing $n$ identical objects among $r$ groups such that each group can have $0$ or more $(\le n)$ objects  I know that standard way of doing this is to solve the problem of distributing n identical objects and $(r-1)$ partitions among themselves which can be done in $C(n+r-1,r-1)$ ways.   But I am unable to prove to myself why it is not $(r+1)^n$.  Because each of the n objects has r+1 choices, either group1, group2,... group r or none at all.
A doubt on the definition of limit of a function definition I want to prove the definition of limit of a function from definition (1) to definition (2).  (1) A function $f(x)$ is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for every sequence $\{x_n\}$ with limit $x^*$, the limit $\lim_{n-&gt;\infty}f(x_n)$ exists and is equal to $f^*$. (2) A function is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for each $\epsilon &gt;0$, there exists a $\delta_\epsilon &gt;0$ such that $|f(x)-f^*| &lt;\epsilon \forall |x^*-x|&lt;\delta_\epsilon$  I am confused about the steps. Suppose I am given an $\epsilon &gt; 0$ and $x^*$. I have to show the existence of a $\delta_\epsilon &gt;0$ such that for all $x$ satisfying  $|x^*-x|&lt;\delta_\epsilon$ I can construct a sequence $x_n$ which converges to $x^*$ and $x_n=x$ for some $n \geq N$. Then I can say that $f(x_n)-f^*=f(x)-f^* &lt; \epsilon$. I think I may have made some mistake about the steps.  My main confusion is the following : Suppose I choose $\delta_\epsilon=\epsilon$. Then if a sequence $\{x_n\}$ converges to $x^*$ then $x_n=x$ (where $x$ satisfies $|x-x^*|&lt;\epsilon$) for some $n\geq N$. Now, $\lim_{n-&gt;\infty}f(x_n)$ exists implies there exists an $M$ such that $\forall n\geq M$, $f(x_n)-f^* &lt;\epsilon$. Now, I need to show that for some $n \geq M$ indeed $x_n=x$. How do I show that ? This $M$ and $N$ are different. IntelliSense not working but it is enabled This problem is really proving to be a tricky one (and quite annoying).  In SQL Server Management Studio 2008, up until a few days ago my IntelliSense was working great.  Then all of a sudden it stopped.  The icon as it as enabled on the toolbar menu, and under Tools -> Options -> Text Editor -> T-SQL -> IntelliSense it says it is enabled there.  I have tried refeshing the IntelliSense cache with Ctrl-Shft-R but that doesn't work either.  Any ideas what happened to my IntelliSense and what I need to do to get it back? Evaluate $$\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}d x$$ What substitution should I use ? $\sqrt\frac{1+x}{1-x}$ suggests $x=cos2\theta $ but its not useful in $\frac{2-3x}{2+3x}$ When & How often do 4e Monster powers recharge? How often should I roll power recharge rolls and at which point in the monster's turn do I re-roll them?  I'm trying to find the official definition of the recharge mechanic in the online compendium and I'm not seeing it there. What was Dumbledore's original wand made of? Will the Garbage Collector call IDisposable.Dispose for me? scanf not taking in data HTML 5 Video stretch The total number of subsets is $2^n$ for $n$ elements In my probability book, it says that to count the total number of subsets of n elements is a process of $n$ stages with binary choice of either adding this element to the subset or not to add it. Therefore, the total number is $$2^n$$  But, for instance, we have 3 elements, according to this formula, there are 2 to the power of 3 elements, namely 8, which are $${\emptyset},A,B,C, AB, AC, BC, ABC$$  However, I have a hard time of imagining the process or N stages binary choice that form this many subsets. Can anyone explain/help me to understand it? I mean, ABC, if we are making the choice of A, put it in or do not put it in, exactly which subset are we choosing to put in or not? Thank you. I have a vf page that is a custom button. That's it.  this VF page is on a STD Page Layout. When the user clicks on the VF page, the parent window needs to be redirected to the new URL.  The way this is working, is that the new page is opening in the vf section of the page layout.   &lt;apex:page standardController="Opportunity"  standardStylesheets="true" extensions="OpportunityController" &gt; &lt;base target="_parent" /&gt; &lt;body style="background:#f3f3ec" &gt; &lt;apex:form &gt; &lt;center&gt;  &lt;apex:commandButton action="{!redirect}"  value="Change Record Type" rendered="{!render}"/&gt; &lt;/center&gt;   &lt;/apex:form&gt;    &lt;/body&gt; &lt;/apex:page&gt; How to Improve Mathematical Thinking and General Problem Solving Skills? I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.  Here's how I generally approach it:   What is the problem? What do I need to do here? Does it look like I've encountered this before?  Can I think of a smaller problem to solve instead?   If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."   What can I do to improve my situation? Search for singleton tags on questions Is there any way to search for questions that have a specific tag and only that lonely tag?  In other words, I'd like to be able to do something like the following to find questions that have a single tag and no other:  [mytag] tagcount:1   This would be very useful for tag cleanup situations to be able to do house-cleaning before nuking a tag. "Screaming" noise while steaming milk? Errors in matrix equation? I'm getting errors in a matrix equation. When I compile it looks fine, but I'd like to fix them so I don't have to press return for every error at compile time. The code for the equation is:  \begin{equation*}  \begin{bmatrix} U(1,t)  &amp; \cdots &amp; U(n,t) \\  \vdots &amp; \ddots &amp; \vdots \\ O_{n1} &amp; \cdots &amp; O_{nn}  \end{bmatrix} \times \begin{bmatrix} W(1,1)  &amp; \cdots &amp; W(1,n) \\  \vdots &amp; \ddots &amp; \vdots \\  W(n,1) &amp; \cdots &amp; W(n,n)  \end{bmatrix} = \begin{bmatrix}  D(1,t)  &amp; \cdots &amp; D(n,t) \\ \vdots &amp; \ddots &amp; \vdots \\ I_{n1} &amp; \cdots &amp; I_{nn} \end{bmatrix}  \end{equation*}   The errors I'm getting are related to missing $'s inserted. How to Avoid Precision Errors While Creating Numpy Arrays? The two causes for the factor 2 in Coriolis effect While reading this document on Coriolis effect , I saw the followig sentence     Two kinematic effects each contribute half of the Coriolis acceleration: relative velocity and the turning of the frame of reference.    And this is the reason why Coriolis term has that factor $2$. Unfortunately it does not specify anything about this two causes. Does anyone have some further explanation for how "relative velocity" and "turning of the frame" actually give rise to the Coriolis term? Is there a limit to a total number of months a holder of UK multiple entry visitor visa can spend in UK per calendar year? A relative of mine has received a multiple-entry UK visa for 2 years. Is there a particular limit on how many months in total per year she can stay in UK? The way I understand the current rule is that such visitors can enter UK as many times as they want as long as they do not stay for longer than 6 months per single visit. However, does this mean that if she stayed with us for 5 months and went back home for 1-2 months, she could come and stay for another 5 months? I heard that it is advisable to spend only 180 day per calendar year in UK and not more, but I could not find that rule in any official UK Home Office documents. Thanks in advance for any info about this. Why is df missing 500MB of available space? I've just had a message today from Ubuntu 11.04 that I have only 100 MB left, so I cleaned up some files, and I got 200 MB. Then, after a couple of hours, suddenly I have only 26 MB?! I tried df, du via mount --bind, /forcefsck with reboot - nothing could should what the culprit is - finally I searched big files, realized /var/log/syslog is 100MB+ and /var/log/kern.log is 200MB+ - blanked them with sudo bash -c 'echo &gt; ...' and rebooted, and now I have some spare MB.  But now, I realize I have another problem with df:  $ df -h Filesystem            Size  Used Avail Use% Mounted on /dev/sda5             9,7G  8,8G  385M  96% / none                  963M  696K  962M   1% /dev none                  969M   12K  969M   1% /dev/shm none                  969M  252K  969M   1% /var/run none                  969M     0  969M   0% /var/lock /dev/sda6             9,7G  8,1G  1,1G  89% /media/disk  $ df -h --block-size M Filesystem           1M-blocks      Used Available Use% Mounted on /dev/sda5                9845M     8960M      385M  96% / none                      963M        1M      962M   1% /dev none                      969M        1M      969M   1% /dev/shm none                      969M        1M      969M   1% /var/run none                      969M        0M      969M   0% /var/lock /dev/sda6                9845M     8235M     1110M  89% /media/disk   Note so for /, it says there are 9845M total; and Used 8960M - then remaining would be 9845-8960 = 885 M, however, here I have only 385M Available.  Also, for /media/disk, it says total 9845M, Used 8235M - then remaining would be 9845-8235 = 1610, however, here I have only 1110M Available.  In both cases, there is exactly 500 MB difference.  Where did this difference come from - and can I reclaim it?  Here is also lsof | grep 'deleted' - can't see nothing suspicious here:  $ lsof | grep 'deleted' nautilus  1911 user   21u      REG        8,5   142760     260562 /home/user/.local/share/gvfs-metadata/home (deleted) nautilus  1911 user   22w      REG        8,5    32768     268807 /home/user/.local/share/gvfs-metadata/home-fe882154.log (deleted) python    1919 user    8u      REG        8,5     4096     392258 /tmp/ffiqRK968 (deleted) python    2165 user    5w      REG        8,5        0     132261 /home/user/.[SNIP].lock (deleted) python    2166 user    5w      REG        8,5        0     132261 /home/user/.[SNIP].lock (deleted) python    2185 user   21r      REG        8,5   142760     260562 /home/user/.local/share/gvfs-metadata/home (deleted) python    2185 user   22r      REG        8,5    32768     268807 /home/user/.local/share/gvfs-metadata/home-fe882154.log (deleted) gnome-ter 2279 user   27u      REG        8,5      640     392575 /tmp/vte5KDX2X (deleted) gnome-ter 2279 user   28u      REG        8,5     4936     392605 /tmp/vteKRDX2X (deleted) gnome-ter 2279 user   29u      REG        8,5      648     392947 /tmp/vteMZDX2X (deleted) ubuntuone 2544 user   17u      REG        8,5     4096     392335 /tmp/ffiMErq0V (deleted) bamfdaemo 3235 user   12r      REG        8,5   143868     269077 /home/user/.local/share/gvfs-metadata/root (deleted) bamfdaemo 3235 user   13r      REG        8,5    32768     272310 /home/user/.local/share/gvfs-metadata/root-18092a02.log (deleted) firefox   5291 user   59u      REG        8,5    33288     132262 /var/tmp/etilqs_YdeZiWSd5iQwJ4U (deleted) firefox   5291 user   60w      REG        8,5    32768     132271 /var/tmp/etilqs_MNLXhEaEqoXMm9b (deleted) firefox   5291 user   70u      REG        8,5    34952     132297 /var/tmp/etilqs_yXDdwVeMxmmdpNz (deleted) How to print original variable's name in Python after it was returned from a function? Too many requests? I went to stackexchange.com and clicked on SO.  After a few minutes I got a message saying "We have detected too many requests from this IP.  If this is an error contact us."  That was the second SE page I tried to visit.  Second only to SE.com  I moved to another machine and it's working now.  The first machine usually works, I don't know if a got assigned a blacklisted IP or what.  Anyone know what's going on?  EDIT: I would really rather not post my IP on the web, but if a diamond or dev asks I'll email it.  EDIT 2: I looked at badp's dupe suggestion and can verify that I have no scripts frequently calling SO.