What are the number of dissimilar terms in the expansion of $(x+\frac{1}{x}+x^2+\frac{1}{x^2})^{15}$?  I know how to solve this kind of problem. First I would arrange the term into a binomial expression. The expansion will have $(n+1)$ dissimilar terms. But how can I arrange it into a binomial expression?
Find the number of dissimilar terms in the expansion of $$\bigg(x+\frac{1}{x}+x^2+\frac{1}{x^2} \bigg)^{20}$$  Could someone give me some hint to proceed in this question. I used $x+1/x=t$ and wrote expansion as $(t^2+t-2)^{20}$ but even this does not convey much information. Please help.
This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so many different ways.  Proof 1 : Euclid's. If there are finitely many primes then $p_1 p_2 ... p_n + 1$ is coprime to all of these guys. This is the basic idea in most proofs : generate a number coprime to all previous primes.  Proof 2 : Consider the sequence $a_n = 2^{2^n} + 1$. We have that  $$ 2^{2^n}-1 = (2^{2^1} - 1) \prod_{m=1}^{n-1} (2^{2^m}+1), $$  so that for $m &lt; n$, $(2^{2^m} + 1, 2^{2^n} + 1) \, | \, (2^{2^n}-1, 2^{2^n} +1) = 1$. Since we have an infinite sequence of numbers coprime in pairs, at least one prime number must divide each one of them and they are all distinct primes, thus giving an infinity of them.  Proof 3 : (Note : I particularly like this one.) Define a topology on $\mathbb Z$ in the following way : a set $\mathscr N$ of integers is said to be open if for every $n \in \mathscr N$ there is an arithmetic progression $\mathscr A$ such that $n \in \mathscr A \subseteq \mathscr N$. This can easily be proven to define a topology on $\mathbb Z$. Note that under this topology arithmetic progressions are open and closed. Supposing there are finitely many primes, notice that this means that the set $$ \mathscr U \,\,\,\, \overset{def}{=} \,\,\, \bigcup_{p} \,\, p \mathbb Z $$ should be open and closed, but by the fundamental theorem of arithmetic, its complement in $\mathbb Z$ is the set $\{ -1, 1 \}$, which is not open, thus giving a contradiction.   Proof 4 : Let $a,b$ be coprime integers and $c &gt; 0$. There exists $x$ such that $(a+bx, c) = 1$. To see this, choose $x$ such that $a+bx \not\equiv 0 \, \mathrm{mod}$ $p_i$ for all primes $p_i$ dividing $c$. If $a \equiv 0 \, \mathrm{mod}$ $p_i$, since $a$ and $b$ are coprime, $b$ has an inverse mod $p_i$, call it $\overline{b}$. Choosing $x \equiv \overline{b} \, \mathrm{mod}$ $p_i$, you are done. If $a \not\equiv 0 \, \mathrm{mod}$ $p_i$, then choosing $x \equiv 0 \, \mathrm{mod}$ $p_i$ works fine. Find $x$ using the Chinese Remainder Theorem.   Now assuming there are finitely many primes, let $c$ be the product of all of them. Our construction generates an integer coprime to $c$, giving a contradiction to the fundamental theorem of arithmetic.  Proof 5 : Dirichlet's theorem on arithmetic progressions (just so that you not bring it up as an example...)  Do you have any other nice proofs? Internal "get/set" function of property? Here is my test code and picture:  import bpy   # Simple property reading/writing from ID properties. # This is what the RNA would do internally. def get_float(self):     return self["testprop"]   def set_float(self, value):     self["testprop"] = value  #bpy.types.Scene.test_float = bpy.props.FloatProperty(get=get_float,set=set_float) bpy.types.Scene.test_float = bpy.props.FloatProperty()   bpy.context.scene.test_float = 12.34     When I don't explicitly set the "get/set" function for my property, after assigning the value, I can get the property value by 'scene["test_float"]', but if I explicitly set the "get/set" function, then if I try to type in the same commands, I will receive a error "KeyError: 'bpy_struct[key]: key "test_float" not found'" . So what should I do to get the same results by using explicitly  "get/set" function? Let's say that you want to output or concat strings. Which of the following styles do you prefer?   var p = new { FirstName = "Bill", LastName = "Gates" }; Console.WriteLine("{0} {1}", p.FirstName, p.LastName); Console.WriteLine(p.FirstName + " " + p.LastName);   Do you rather use format or do you simply concat strings? What is your favorite? Is one of these hurting your eyes?  Do you have any rational arguments to use one and not the other?  I'd go for the second one. How to use an existing database with an Android application Community Wiki posts don't change usernames if the user does Using methods from another class in the main? How many stickers do I need to complete my FIFA Panini album? I'm playing the , which is an Internet adaption of the classic Panini albums that are usually published for the soccer world cup, European championship, and possibly other tournaments.  The album has placeholders for 424 different stickers. The purpose of the game is to collect all 424. The stickers come in packs of 5, which can be obtained through codes found online (or, in case of the classic printed album, bought at your local newsstand).  I make the following assumptions:   All stickers are published in the same quantity.  One pack of stickers does not contain duplicates.     How can I find out how many packs of stickers I need to acquire in order to be reasonably sure (let's say 90%) that I have all 424 unique stickers? How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. Open a direct file on the hard drive from Firefox (file:///) I have a small PHP system I created for myself. This system contains a few .doc and .docx file documents. I want to be able to open them directly from the browser without downloading them.  I created a link: &lt;a href="file:///c:/mysite/myfile.docx"&gt;myfile&lt;/a&gt;. But Firefox does not open the link. href="http://localhost/myfile.docx" works fine - it offers me to download the file.  Any ideas on how I can do it? Do I need to change something in the Firefox options or security settings? $\sin(n)$ subsequence limits set We were given a challenge by our calculus professor and I've been stuck on it for a while now.   Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).  What would be a good way to start? How can I set default value in ActiveRecord?  I see a post from Pratik that describes an ugly, complicated chunk of code:   class Item &lt; ActiveRecord::Base     def initialize_with_defaults(attrs = nil, &amp;block)     initialize_without_defaults(attrs) do       setter = lambda { |key, value| self.send("#{key.to_s}=", value) unless         !attrs.nil? &amp;&amp; attrs.keys.map(&amp;:to_s).include?(key.to_s) }       setter.call('scheduler_type', 'hotseat')       yield self if block_given?     end   end   alias_method_chain :initialize, :defaults end   I have seen the following examples googling around:    def initialize      super     self.status = ACTIVE unless self.status   end   and    def after_initialize      return unless new_record?     self.status = ACTIVE   end   I've also seen people put it in their migration, but I'd rather see it defined in the model code.  Is there a canonical way to set default value for fields in ActiveRecord model? Communicating with all potential clients as a multi niche photographer This may be seen as a bit off topic, but I'm hoping it's a question that some photographers can associalte with and clear up for me.  I'm an amateur wanting to get more serious in the new year, and aiming to build up a portfolio of work that I will showcase on my own website, which will hopefully build up a client base.  I figured that a good way to get started would be to create a few 365 style projects that I can publish on my site and on social media. However, my interests vary a great deal, and therefore my photography will more than likely range from nature and local events, to more NSFW style topics.  My problem is, how to successfully communicate with people who will be interested in certain aspects of my work, without losing touch with those interested in others.  In theory, I'd like to set up a single social media presence, preferably Twitter, where I can freely post anything I want, and connect with anyone I want. Essentailly pointing people in the direction of whatever content they will be interested in. Problem is, social media sites like Twitter publicise ALL of your posts to ALL of your followers, so people who are following for nature/local photography will possibly unfollow me for my NSFW content. At the same time, some SFW followers may be potential clients in other areas, and vice versa.  So as a multi niche photographer, how can you engage with all potential clients, and showcase all portfolio peices ( in my case all 365 project shots ) via a single or limited number of social networks? How are you doing it? Why does a helium filled balloon move forward in a car when the car is accelerating? I noticed that when I had a helium filled, latex balloon inside of my car, it moved forward in the cabin as I accelerated forward. The faster I accelerated forward, the faster the balloon went from the back of the car to the front of the car. The balloon didn't have a string.  This became a game with my 4 year old as we drove home. We figured out where the balloon would go based on how fast I accelerated, turned corners etc.  I expected that it would act a lot like the water in a cup does, but it was the total opposite it seemed.  What forces caused this behavior?  I assumed it has something to do with the fluid dynamics in the closed cabin, but I can't figure it out. How do I wrap text in a pre tag? I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$   How do I proceed I have calculated $$4 E\frac{(X-\theta)^2}{1+X^2+\theta^2-2X\theta}$$  Can somebody help How does a wheelie work? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. Last minute flight bookings - better to book online or at the airport? We had , which covered ground on what happens to flight prices over the period.  What I'm interested to know is if I had to fly out at short notice, say within 1-2 days, then am I better off booking within that 1-2 day gap online, or am I better off trying my luck directly at the airport (as long as I'm willing to wait at the airport) for a cheaper price? Is there any truth to rumours that airlines sell tickets at cheaper prices at the airport right before flights to fill empty seats or is that a myth? While I'm aware of  it does not quite answer my question. I have a simple sentence:     If any of the devices was not switched off, do something.   Is there supposed to be was or were? I was thinking about was since it could be rephrased as:     If any one of the devices was not switched off, do something.   But maybe that does not make any sense at all. Also, this is for a technical documentation where more formal language is preferred, not sure if that makes any difference. Is the integral $\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$ convergent? MBP 13' (mid-2010; 7,1) 2.4 Core 2 Duo P8600 supporting 16 GB RAM under Win7? I’ve got a MBP 13′ (mid-2010) 2.4 Core 2 Duo (full specs below) that I’m now operating solely under Win7 64Bit. It formerly run as well under OS X 10.6.8, relatively to which the firmware seems up to date (MBP71.0039.B0B).  Would my system recognize 16 GB RAM if operated under Win7 64 Bit? Or would it require the latest firmware (MBP71.0039.B0E) of OS-X 10.7.5 (which was never installed on my MBP) anyway?   As far as I have read elsewhere, using 16 GB RAM under OS X seems to definitely require this later version of OS X (including the respective firmware update); so I'd like to know whether this is an issue of OS X or (as I suspect) the firmware.    Modellname: MacBook Pro Modell-Identifizierung: MacBookPro7,1 Prozessortyp: Intel Core 2 Duo Prozessorgeschwindigkeit: 2,4 GHz Anzahl der Prozessoren: 1 Gesamtzahl der Kerne: 2 L2-Cache: 3 MB Speicher: 4 GB Busgeschwindigkeit: 1,07 GHz Boot-ROM-Version: MBP71.0039.B0B SMC-Version (System): 1.62f6