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What is the coefficient of the term $a^i b^j c^k$ in the expansion of $(a+b+c)^n$? What is the coefficient of the term $a^i b^j c^k$  in the expansion of   $(a+b+c)^n$ ?   Given solution:     Before collecting like terms, each term in the expansion is a product   of $n$ factors each  of which is $a, b,$ or $c$.  Those terms which   have exactly i a’s, j b’s and k c’s can be combined into   the single term $a^i b^j c^k$ and the number of such terms will be   equal to the number of n letter strings containing i a’s, j b’s and k   c’s, which by the Mississippi rule is $\frac{n!}{(i!\times j! \times  k!)}$.   I understand that this problem is actually about the derivation of the coefficient of the multinomial theorem,but I am not sure about the combinatoric reasoning,what I mean is that I understand the analogy problem but what I couldn't is how the analogy holds here,Could anybody explain this to me? I read this sentence in a book:     I just took part in a study over in the Psychology Department.   Why do we add over in front of in the here? When do I have to use 'will' instead of 'going to'? Does going to only express an intention and will some kind of prediction that doesn't necessarily happen?  EDIT: Thanks for all your answers. I asked this question because I always fail to complete exercises like this from an ESL book:   We  will fly to Venice in June. (Correct: But, how can I be sure that I'm not sick in June?) Philipp will be 15 next Wednesday (Correct: Since his birthday is a fixed day, this will happen). They will get a new computer. (Wrong: Why?, If I need a new computer, I simply get a new one) In 2020 people are going to buy more hybrid cars.  (Wrong, Why? Who can tell for sure). Some English words only differ in their vowels: crisscross, dillydally, riffraff, etc. Is there a name for them? Can I use a one time pad key twice with random plaintext? Intuition about Herbrand Models. I'm having trouble understanding how Herbrand models work. I feel like there is a bigger picture to it which I'm not understanding. Why does every satisfiable first order formula have a herbrand model at all ? Can I somehow construct a Herbrand model from a normal model ? My book also says that : a formula is unsatisfiable iff the Herbrand expansion is unsatisfiable. The book I'm using &quot;Mathematical Logic for computer science by Mordechai Ben-Ari&quot; has all the proofs of the above but its not very intuitive to me. Could somebody share their point of view? VB6 compiling via console (vb.exe) compatibility issue on 64bit (works on 32bit?) Replacement for alliums? Opt in "mentor" privilege at 7.5k I am suggesting a Mentor privilege at 7.5k. It would be on a per-user per-site opt in basis.  At some reputation level (perhaps 7.5k) there could be an opt in "mentor" privilege.  If someone opts in then when a new user signs up, they become their Mentor. The new user is the Mentee.  They will get notified of any posts or suggested edits they make, and can help and support them.  While this would by no means help everyone (I'm sure there are more new people signing up than there are 7.5k people willing to help them), it could help mitigate the "new user's bad posts" and such like.  By no means would being someone's mentor mean you are the only person who can edit and help them, and you aren't obliged to do so - but if you have opted in then you should help.  How would this work?   Any Mentor would have no more than 5 active Mentees at once. Once the Mentee gets to 125 rep, the downvote privilege (or any other level deemed sensible) they are no longer a Mentee. If they aren't active for a month, they are no longer a Mentee. If the Mentor has 5 Mentees (i.e. cannot mentor anyone else) then any Mentee not active for 2 weeks is no longer a Mentee. The Mentee can opt out at any time, the Mentor can opt out, but if they opt out a lot (say 5 people in 2 months) they are blocked from the privilege for 2 weeks. The Mentor gets no extra rep. They can still get rep from answering their Mentee's questions and similar. Once the new user gets 20 rep on the site, a chat room is created for them to talk. No user with the association bonus will become a Mentee. This would (maybe) be opt in for sites. For example, this might not be at all useful for this site, or for , but on  or  I imagine this could be quite good.    appears to be asking the same thing, based on the title, but that is asking about people who know each other offline mentoring each other.    Every single one of these suggestions (especially the numbers) is open to discussion. Maybe 125 is to low to stop being mentored, or maybe it's too high. What do you think? I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that.  The author defines an embedding between two manifolds $f: N\rightarrow M$ as an injective immersion (differential is injective everywhere) with the additional property that $f(N)$ as a subspace of $M$ is homeomorphic via $f$ to $N$. Then a canonical form theorem is proven, namely, that for any point $p \in N$, there are charts $(U,\phi = x^1, \ldots, x^n) \ni p$ and $(V,\psi = y^1, \ldots, y^m) \ni f(p)$, such that the transition map takes the following form:  $$\psi \circ f \circ \phi^{-1} : (r^1,\ldots,r^n) \mapsto (r^1,\ldots,r^n,0,\ldots,0).$$  (I guess it is implicitly assumed that $f(U) \subset V$).  Now in the process of proving that $f(N) \subset M$ is an , he states that $f(U) \subset V$ is defined by the vanishing of the coordinates $y^{n+1},\ldots,y^m$. I can prove that $$ f(U) \subset \{q \in V: y^{n+1}(q) = \cdots = y^m(q)\} $$ but not the reverse inclusion. Can anyone explain why this should be true? Using THE before some countries I have been taught that it's grammatically proper to use the before countries like "Sudan and Yemen" but I was never given a reason (if there exists any, of course) when I asked why, except for "it is one of those ambiguous parts of English language that we &ndash; as non-native speakers &ndash; should just accept without trying to look for a reason". Now, I'd like to know if there is any answer to this "why" or I should keep on believing that there are no hard and fast rules when it comes to questions like these. "There Is"/"There are" depends on plurality of the first list element or not? It seems I put a   Bounty assigned by outside party, two lengthy, reference-citing answers, one "-1" (awarded the bounty), one "-2", two others scored "0" and "-2" respectively, the answers suggesting one or the other is correct, 73 comments and no consensus so far - and me, as the asker, lost without a clue what to think of the answers anymore -  this is no longer "Learners' Level" question, so I thought I'd bring it here and hear what the experts have to say.    Original question:  Plurality of verb depending on plurality of list elements  An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that - was my editor overzealous or am I trying to copy rules of my language that don't apply in English?    Someone linked a  for negatives, where the situation is more clear-cut ("There is no..."). Same applies to connection with "or" apparently, per accepted answer there. Still, nothing for "and".  It seems there is a consensus that in that if the verb goes after the list, it will be plural.     an orange, some grapes, two apples and a small pile of cherries were on the plate.   There doesn't seem to be clear consensus for:     On the plate { was | were } an orange, some grapes, two apples and a small pile of cherries   Adding more to the confusion is the abbreviated form: "There's"     One last note. In North America, at least, there is a widespread use of using a singular form like "there is" and "there was", without regard for the subject item or items, and this "there is" is often shortened to "there's":           There's three apples on the table!      Could you please clarify this mess? How to install Windows 7 after Ubuntu and dual boot? I just installed Ubuntu, a full install removing the prior OS. I was forced to remove  from my  because of the max four partitions already being there. I didn't lose anything, all backed up. However, now I am looking to create another partition and install Windows&nbsp;7 on it.   Do I also need some sort of boot loader that lets me choose which to boot? I'm making a door with piston sliding sideways that reacts to pressure plates. As the pressure plates are sending the current to redstone at a very low level, I need to raise it somehow, and this leads to awkward stairs of redstone.    So I'm wondering whether there are better ways of sending the redstone current up, maybe even vertically. Why do electron and proton have the same but opposite electric charge? Multivariate polynomial with no mixed terms Is there a standard name for multivariate polynomials wherein each term consists of only one coordinate? That is, polynomials of this form: $$p(x_1, \ldots, x_n)= \sum_{i=1}^n \sum_{j=1}^n a_{i,j}x_i^{j}$$ where all of the $a_{i,j}$ are constants. How do I call a .NET assembly from C/C++? Suppose I am writing an application in C++ and C#.  I want to write the low level parts in C++ and write the high level logic in C#.  How can I load a .NET assembly from my C++ program and start calling methods and accessing the properties of my C# classes? I was running my OS off a USB stick fine and the wifi was working. I installed it to my SSD and setup again but not the WIFI adaptor isn't working.  I followed the first time  and it worked but since installing onto the SSD and doing the same its not working.  No wlan0 interface found Please check drivers, try lsusb to check USB device is present root@goldminer:~# lsusb Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 002: ID 0bda:0811 Realtek Semiconductor Corp.  Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub   On the working device  root@coalminer:~# dkms status amdgpu, 18.10-572953, 4.13.16-hiveos, x86_64: installed amdgpu, 18.10-572953, 4.4.0-127-generic, x86_64: installed nvidia, 390.59, 4.13.16-hiveos, x86_64: installed nvidia, 390.59, 4.4.0-127-generic, x86_64: installed rtl8812au, 4.2.2, 4.13.16-hiveos, x86_64: installed rtl8812au, 4.2.2, 4.4.0-127-generic, x86_64: installed   On the SSD  root@goldminer:~# dkms status amdgpu, 18.10-572953, 4.13.16-hiveos, x86_64: installed amdgpu, 18.10-572953, 4.4.0-127-generic, x86_64: installed nvidia, 390.59, 4.13.16-hiveos, x86_64: installed nvidia, 390.59, 4.4.0-127-generic, x86_64: installed rtl8812au, 4.2.2, 4.13.16-hiveos, x86_64: installed (WARNING! Diff between built and installed module!) rtl8812au, 4.3.8.12175.20140902+dfsg: added rtl8812au, 5.2.9: added   Any advise? I think from trying to fix it ive been making it worse. Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$? I can solve this equation $x(x-1)(x-2)(x-3)=1$ using the usual method but I am looking for a fast analytical method to solve this. Any hints ? How to create animated GIF images of a screencast? I've seen animated GIF images of screen casts (like the one below) promoted a few times on this site as a way to improve answers.    What toolchain is being used to create these? Is there a program that does this automagically, or are people taking screencasts, converting them into a series of static frames, and then creating the GIF images?