The number of ways of distributing $n$ identical items among $r$ people such that each one of them receives at least one is $\binom{n-1}{r-1}$. Can you please give a simple proof to explain how is this theorem derived ? Thanks .
I am trying to understand the derivation of the stars and bars formula. I understand how a problem can be converted to the stars and bars format, but I am confused about how the problem of choosing all the different ways to put k stars in n sections can be written as ${n+k-1 \choose k}$. Generally "n choose k" is for counting the number of combinations of k objects we can get given n distinct objects right? That I understand since we're doing permutation divided by the number of orderings for each group since order doesn't matter. I need help figuring out how stars and bars fits into this.
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small? I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them. I'm new to regular expressions and would appreciate your help.  I'm trying to put together an expression that will split the example string using all spaces that are not surrounded by single or double quotes. My last attempt looks like this: (?!") and isn't quite working. It's splitting on the space before the quote.  Example input:   This is a string that "will be" highlighted when your 'regular expression' matches something.   Desired output:  This is a string that will be highlighted when your regular expression matches something.   Note that "will be" and 'regular expression' retain the space between the words. Hope someone can answer my Question. I live in Milwaukee WI.My neighbor just informed me she will have to have her basement excavated. The side of her house is 14 feet away from the side of my house. I have a bi-level home. On the side of my home is a finished family room and bedroom. Should I be worried about heavy machinery doing damage to the foundation to my home that could cause problems in the future. Thank You in advance of your answers. I am trying to show that if for a group $G$ we have $Z(G)=1$, then $Z(\operatorname{Aut}(G))=1$.  I have tried everything I can think of, but to no avail.  The first part of the problem was to show that $\operatorname{Inn}(G) \triangleleft  \operatorname{Aut}(G)$ - does this have any connection?  I couldn't find one.  I know that $\operatorname{Inn}(G) \cong G/Z(G)$, so in this case $\operatorname{Inn}(G) \cong G$ and $Z(\operatorname{Inn}(G))=1$.  Is there any way to get from this to $Z(\operatorname{Aut}(G))=1$?  How do we go about showing this?  Thanks. how to have a custom zsh auto_cd command Instead of the normal cd I've created my own function  function cd {     builtin cd "$@" &amp;&amp; ls -F }   This works great for when I use commands like cd someDir/subDir but not when I change directories via zsh's auto_cd someDir/subDir.  Is there a way to customize what command gets called for auto_cd? Does the Rule of Two Serve any Actual Purpose? I know Darth Bane came up with the Rule of Two to ensure the survival of the Sith, but, in reality (or in the Star Wars reality), does it (or any Sith rule) serve a true purpose?  And why does each successive Sith bother with it?  Due to the nature of the Sith, Sith Lords tend toward arrogance and self-importance.  Both the Sith master and Sith apprentice seem to regularly break the rule by training others.  Deceit and personal interest over loyalty is a way of life for the Sith.  What is to keep any Sith from following the Rule of Two (or any rule, for that matter) if it is against their own self interest?  If a Sith Master can stay in power longer by training a 2nd apprentice and pitting the two against each other, is he really going to worry about the Rule of Two more than his own plans and survival?  The Rule of Two also assumes that every Sith Master would place the survival of the Sith over his own survival, as opposed to attempting to ensure his own immortality in whatever way he could (like Plagueis).  When you have a group that, by default, is arrogant and self-serving, as well as deceptive, why would they want the Sith to survive themselves?  I can see how the Rule of Two, and any other Sith beliefs would be possible guidelines, but it terms of the nature of ambitious people (and the Sith are ambitious), it's hard to believe that each new Sith Master would follow this rule faithfully.  So does this rule, or any Sith rule, actually work?  Or is it more of a suggestion that is broken when convenient? If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.   I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue. Can anybody give me a hint? Can a service written in .NET self-terminate? I have a service application written in C# and under certain circumstances, I would like it to terminate itself.  This would happen after the service has been running for a while, so this would not be happening in the OnStart() event.  Everything that I have read so far suggests that the only safe way to terminate a service is through the Service Control Manager.  My service runs as Local Service and does not have the rights to start or stop services, so I can't access the SCM from the service itself.  Is there another way to self-terminate while still playing by the rules of the SCM? How to tell if a certain item is in a player inventory? I am trying to create a challenge world where a certain item (e.g. a Netherite Sword) must be collected. How can I check automatically if a player has it (using commands), and then congratulate the winner? This question is specific to Bedrock Edition 1.16. I have tried other solutions so many times and they don't seem to work. See also: I have 10 years of daily returns data for 28 different currencies. I wish to extract the first principal component, but rather than operate PCA on the whole 10 years, I want to rollapply a 2 year window, because the currencies' behaviours evolve and so I wish to reflect this. However I have a major problem, that is that both the princomp() and prcomp() functions will often jump from positive to negative loadings in adjacent PCA analyses (ie 1 day apart). Have a look at the loading chart for the EUR currency:    Clearly I can't use this because adjacent loadings will jump from positive to negative, so my series which uses them will be erroneous. Now take a look at the absolute value of the EUR currency loading:    The problem is of course that I still cannot use this because you can see from the top chart that the loading does go from negative to positive and back at times, a characteristic which I need to preserve.   Is there any way I can get around this problem? Can I force the eigenvector orientation to always be the same in adjacent PCAs?   By the way this problem also occurs with the FactoMineR PCA() function. The code for the rollapply is here:  rollapply(retmat, windowl, function(x) summary(princomp(x))$loadings[, 1], by.column = FALSE, align = "right") -&gt; princomproll No, this is not a , nor is it a solicitation on  to backup or  and only a little bit about Where to backup and When to do it.  This is basically a strategy question: what, where and when all together. "How" is not really relevant: there are tons of back-up programs out there and they all do the same basic thing: make copies of data.  But the real question is: are these back-up programs making copies of the data important to you?  How should you install&amp;run Ubuntu while safeguarding your data?  There is no "one size fits all" when it comes to computers, Ubuntu versions or a backup strategy, therefore I've split this up into 5 basic user types with each their own answer:       (or you're running in UEFI mode!)   And as the above does not fit everyone, one for the impatient:   !    So click one of the above links and go to your user type!  P.S. If you're running a server farm and you've read this question so far, I'm sorry: this is about a single PC, not a network, nor a server farm. Is it possible to run Drush without root? I run a Linux server that is administered by a friend of mine, and at the moment I work with Filezilla.  I've heard that Drush is a trivial installation: it seems to require no configuration and work out of the box.   However, it looks for configuration in several places:     Drush installation directory,  the home directory (as .drushrc.php),    root directory of a Drupal installation, and  site directory of a   Drupal installation.  Since the configuration files are just PHP   scripts, they can  contain any PHP code that you would like to have   executed on startup of Drush.  one could for instance call ini_set()   to set the value of a PHP configuration option.   and the configuration file /opt/drush/drushrc.php with following content:    # Allow Drush to use 128 MB of memory. Use -1 for no limit. ini_set('memory_limit', '128M');   I see this is necessary because PHP's default is too low, but I'm not sure if I need higher-level privileges because of it.  Do I have to have root-level permissions on the server, or am I able to work with Drush without them?   Update:  As far as i understand this, i only have to ask my admin to get some more privileges (permissions)Note according to BetaRides answer i need to have more than only the access-data that the FileZilla needs - that are provided for the filezilla! Guess so - but i don not need root-permissions.  btw - i have secure-shell-access (ssh2 access) guess that this is enough -With that i can setup a sftp-connection to the server with filezilla just lemme know - if i can use the exactly same data for the shell access (with the command line tool=?!= How can the Xenomorph see? If you just look at a picture of one there are no eyes readily apparent but they clearly know where they are going. Could their screeching be some form of echo-location? What is this "forbidden" icon in my systray? The following is a systray applet icon that appeared in my Unity session after I upgraded to Ubuntu 16.10. Its menu is tall, narrow and empty. What is it? No file output after rendering? I usually am able to render successfully to the c:\tmp folder, but when I came into work today, I rendered out the image on my work laptop, and everything seems to work, but I'm not getting an output to the tmp directory.  Does anyone know how to make this work? Comparing integers: arithmetic expression or conditional expression In Bash, two integers can be compared using conditional expression     arg1 OP arg2      OP is one of -eq, -ne, -lt, -le, -gt, or -ge. These   arithmetic binary operators return true if arg1 is equal to, not equal   to, less than, less than or equal to, greater than, or greater than or   equal to arg2, respectively. Arg1 and arg2 may be positive or negative   integers.   or arithmetic expression:     &lt;= &gt;= &lt; &gt; comparison      == != equality and inequality   Why do we have two different ways for comparing two integers? When to use which?  For example, [[ 3 -lt 2 ]] uses conditional expression, and (( 3 &lt; 2 )) uses arithmetic expression. Both returns 0 when the comparison is true  When comparing two integers, can these two methods always be used interchangeably? If yes, why does Bash have two methods rather than one? Testing stored procedure scalability I have an email application that will be called upon to deliver to the UI the number of new messages for a given user on each page load. I have a few variations of things I am testing on the DB level but all are abstracted by the stored proc call.  I'm trying to slam the DB to see what the breaking point (# of requests per second) would be.   In a nutshell, I have a table such as this userId, newMsgCount with a clustered index on userId. SQL should be able to server hundreds or thousands of these responses per second. I think the laggard is my .NET app.  How can I make this a good test to achieve the test results based on SQL performance?  Is there a tool for this that i can give it a stored proc name and param for it to pund my  DB?  I want to see if the DB can return a min. of 250 responses per second. There are many collectable flags throughout the game of Assassin's Creed and every time one is collected, you get a progress indicator (e.g., 33 of 100).  Is there any reason to collect these, other than bragging rights?  I recall getting all available flags in one particular town and never saw any special reward or secret revealed. Is there a way to call process builder from Apex?