Find the value of this combinatorial sum. $\sum_{k=4}^{100}\binom{k-1}{3}$
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
I need to prove that the language $\mathscr L=\{0^n \big|\text{n is the square of a natural number}\}$ is non regular using the pumping lemma   My try:  $\mathscr L=\{\overbrace{\epsilon}^{0^2},\overbrace{0}^{1^2},\overbrace{0000}^{2^2},\overbrace{000000000}^{3^2},\overbrace{0000000000000000}^{4^2},\dots\}$  Suppose that $\mathscr L$ is regular so $\exists$ a word '$x$' with length of at least $n$   $|x|\geq n $ such that  $(1)\,\,\,|uv|\leq n$  $(2)\,\,\,|v|\geq 1$  $(3)\,\,\,uv^iw \in \mathscr L$  Now, let as choose the word $\color{blue}{x=0^{n^2}}$  $|x|\geq n$ so we can use $(1)-(3)$  $x=\underbrace{000000\dots}_{n^2\text{ times}}$  $x=uv0^kw=\underbrace{uv0^k}_{\text{n zeros   }}\overbrace{w}^{n^2-n\text{ zeros }}$  Let as choose $\color{blue}{i=??}$      I'm difficult choosing the word '$\color{blue}x$' and choosing  $\color{blue}i$ to come to contradiction with the lemma This is a  about Active Directory domain naming.   After experimenting with Windows domains and domain controllers in a virtual environment, I've realized that having an active directory domain named identically to a DNS domain is bad idea (Meaning that having example.com as an Active Directory name is no good when we have the example.com domain name registered for use as our website).   , but I'm still not sure about what other rules there are around naming Active Directory domains.  Are there any best practices on what an Active Directory name should or shouldn't be? Is there a preferred way to return multiple values from a C++ function?  For example, imagine a function that divides two integers and returns both the quotient and the remainder.  One way I commonly see is to use reference parameters:  void divide(int dividend, int divisor, int&amp; quotient, int&amp; remainder);   A variation is to return one value and pass the other through a reference parameter:  int divide(int dividend, int divisor, int&amp; remainder);   Another way would be to declare a struct to contain all of the results and return that:  struct divide_result {     int quotient;     int remainder; };  divide_result divide(int dividend, int divisor);   Is one of these ways generally preferred, or are there other suggestions?  Edit: In the real-world code, there may be more than two results.  They may also be of different types. Edit a truetype font I'm trying to find a free tool (on windows ) that would enable me to change some settings on a TrueType font. Specifically, I'm trying to fix the following errors:  E1311 The macStyle bold bit doesn't match the name table's font subfamily string macStyle bold bit is set, but subfamily is Regular  W2000 The subfamily string is inconsistent with the style of the font OS/2.fsSelection bold bit is set, but subfamily string = 'Regular'  I can do that with fontographer, but would prefer to use something free.  Fontforge doesn't seem to work with this font, when saved it corrupts the cyrillic letters in the font. Real UID, effective UID, and some systems even have a "saved UID". What's the purpose of all these, especially the last one? I know a lot of female programmers, and I know there are a good number of them out there.  But I don't recall ever having one of my questions answered by, nor have I ever answered a question by a female programmer here at Stack&nbsp;Overflow.  Now I mostly deal in the Delphi tag, and that may make a difference. But I've got the feeling there is something about the programming social network that Stack&nbsp;Overflow is, along with its form of mini-competition (via reputation) that turns most ladies off.  When I go down the first few pages of top users by rep, there is nary a lady to be found.  Go over to Quantcast and . Go down to the Demographics, and you'll see that a full 20% of the visits are by females. So they are here, lurking and using the wonderful info that Stack&nbsp;Overflow provides.  Then why aren't there more of them participating, both with questions and answers?  What can Stack&nbsp;Overflow do to persuade female programmers to participate more?    So after a dozen downvotes offset by a dozen upvotes, and accusations by many that I am a bigot, the bottom line seems to be that Stack&nbsp;Overflow appeals to males more than females.  This is my observation and I believe a proper analysis would support the hypotheses that a much higher percentage of male programmers use Stack&nbsp;Overflow than the percentage of female programmers.    Assuming that is the case, then Stack&nbsp;Overflow is geared more towards the male psyche than the female. If so, I again ask, what can they do about that?      This question was asked in StackOverflow Meta on November 20, 2009. It was made Community Wiki. It was asked in a constructive manner and should be considered an appropriate question for Meta.   Yet it was closed over two years after it was asked as "not constructive" after more than 40 upvotes and 10 favorites and 16 answers.   Now it resides here on Meta StackExchange, still closed. I just shake my head. Why is the observable universe so big? List all files in all subfolders How to remove symbolic link I was trying to create this symbolic link:  sudo ln -s /usr/share/phpmyadmin /var/www/phpmyadmin   but I accidentally typed:  sudo ln -s /usr/share/php,yad,in /var/www/phpmyadmin   So now I want to correct it but it says symbolic link already exist. How do I enable or disable the global application menu? When for the first time A new question I answer This haiku appears:     Little slab of meat   In a wash of clear jelly   Now I heat the pan   Such a message is Very strange; is anyone Seeing the same thing?  Have not seen again. Came and went like the wind does! Why I am so blessed? An example:    O  O?   There is a triple letter tile to the right of the bottom O, indicated with a ?.  If I place an X on it, making the words OX (vertical) and OX (horizontal), do I score 50 points (scoring the tile twice) or 34 (scoring it once)? Silencing "Your disk is almost full" notification After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? Vertical line of white pixels between launcher and maximized application (Firefox) Is this a bug or is it expected?  How can it be solved if it's not expected? Historical reason behind different line ending at different platforms Why did DOS/Windows and Mac decide to use \r\n and \r for line ending instead of \n? Was it just a result of trying to be "different" from Unix?  And now that Mac OS X is Unix (-like), did Apple switch to \n from \r? Correct spelling of Polish surname Dziubenofska? My Polish grandmothers last name was Dziubenofska, but I know that's not how it is spelt! I'm looking for the proper spelling.   Can you help find it for me?  She came over to Ellis Island and they changed it on her to dziuk. This is only a prefix to the full name. Security importance of Key Schedule in Block Cipher In Harry Potter, Do Spells Travel in a Straight Line? In the Harry Potter world, when a magical person casts a spell, sometimes they target a specific object, like when Harry has a wand only and needs his broom, he casts "Accio Firebolt" and his broom comes to him.  But a lot of the time people point their wands just in the direction of someone to cast a spell.  One example would be in a duel.  When they do this, are they casting a spell on a particular person or at that person?  It seems, from what I remember of the movies, that some spells travel in a straight line while others specifically target a person or object.  Do any spells travel in one direction, like a laser beam?  What kind of spells are directional (instead of focusing on a particular person or object)?  And can directional spells be deflected or blocked by something like a mirror or another physical object?  (By "directional," I could have said linear, but didn't want to start sounding mathematical or scientific.  In other words, do they travel in one direction, like a laser beam would, or will they change direction to hit the intended target?) Is there any bidirectional 5v-3.3v level shifter? Is there any IC that interfaces any 3.3V Input/Output to 5V Output/Input? I need it primarily for an Arduino Due but if there any bidirectional IC that works like that it would be great.  Some people advised me to use the SN74AHC125 and CD4050 ICs, but I don't understand how they work or how to interface with them. $2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$