Let $V_m$ be the vector space of polynomials in $k[x_1,...,x_n]$ with degree at most $m$. Then what is the dimension of this vector space?   I can do this with m&lt;4 maybe. But what is the general formula? It seems that this is a combinatorics problem. But I really do not have idea how to find a formula.
says that the number of different monomials of degree $M$ in $N$ variables is $$\frac{(M+N-1)!}{M!(N-1)!}\; .$$ Can anyone explain why this is true?
What is the name of the problem where lines appear in a photo where there should just be a smooth gradient? Every partial order can be extended to a linear ordering How do I show that every partial order can be extended to a linear ordering?      I think that I manage to prove that claim for finite set, how can I prove it for infinite set?  Thank you. command not found and syntax error near unexpected token Movie: Creature crawls out of its current hosts mouth into next hosts mouth I did a bit of math at school and it seems like an easy one - what am I missing?  $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$  $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ 0}_{n\text{ times}} = 0$$  (i.e add $0$ to $0$ as many times as you like, result is $0$)  So I thought an infinite number of $0$'s cannot be anything but $0$? But someone claims different but couldn't offer a reasonable explanation why. Google results seemed a bit iffy on the subject - hopefully this question will change that. I'm trying to test a page that has an input but really its a drop down list. The user enters text and then a dropdown appears based on what the user inputted. They are groups that are in the database. If I look for these elements before typing anything they dont exist but once I start typing they do exist.  The problem is I cant just type the whole name and then press enter or return because an option has to be selected from the list.  I can press arrow down to get the focus on the item from the list but then if I say enter it doesnt select, it sends that enter to the original element and nothing gets selected.  I can find the select element:  var ddList = driver.FindElement(By.CssSelector(".ui-autocomplete-input")); ddList.SendKeys("Admins");   This is the first time the list appears in the html.  It looks like this:  &lt;li class="ui-menu-item" role="presentation"&gt; How do you use plural or singular form of an adjective? In my Computer science class I was given a problem where I have to simulate a bouncing ball using "real physics". I have been trying to find a equation that will simulate the height of the bounce given a gravity and an arbitrary mass. And I will need to calculate the next bounce and it's height. A lot of the equations I've found require a time. But I don't really care about a time, all I want is to get the next height after the previous bounce until it finally hits height of 0 or close to it. I haven't taken a physics class since high school( 5 years ago) and that was basic physics. Does copying a spell from a scroll into your spellbook consume the scroll? The text on page 32 of the Basic Rules that talks about copying spells does not explicitly state that the process consumes the scroll. Does it? How can I query SEDE for tags I've created? How can I query the  for tags I've created? Exponential of the differential operator I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and  $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$  (ref. )  So I was curious as to whether the differential equation  $$\frac{df(t)}{dt} = e^Df(t) = f(t+1)$$  has any solutions apart from $f = 0$? How to make a multiboot CD that will start a user-chosen ISO file I have many CD ISOs downloaded from the Internet. They are each about 100 MB in size and consist of repair/diagnostic programs and other tools. Some of them are Linux-based; others are based on DOS clones, like FreeDOS. I want to put all of the ISOs onto one bootable DVD, so that when I use the DVD, I will get a menu prompting me to select the ISO I want to boot. How can I do this? I've seen some rescue CDs that are combinations of other people's ISOs, but I have no idea how they were made.  Regarding bounty: I, a different person than the OP, have placed a bounty on this question for a solution that meets ALL of these criteria:  Is for DVDs (not flash drives!!!) Is free Works with any ISO file, regardless of original OS I'm trying to wrap my head around this very Hello World-y problem in game development. I've created a TicTacToe game in XNA so I guess the next step would be a  clone.  Keep in mind that I have no knowledge on game programming or even what maths I should apply to where. That's why I'm asking this question.    To the question: How can I determine where the ball should bounce when it hits the paddle at the bottom of the screen?  I'd imagine it would be something like:   Capture speed and angle from incoming ball. Detect where it touched the bar (far left, far right, center) and according to that give it a higher speed if it touched the outer areas. This is where I'm stuck. Hehe.   Any insights? I realize this is not a straightforward answer type question, but I'm sure it's one everyone faces at some point.  I'm reading the book Linear Algebra that was recommended on this website, but I still have no idea if I should apply it here. How can I make a deepcopy of a function in Python? Case insensitive 'Contains(string)' What techniques have been used to repeat geometry or other detail to the infinite horizon? When I composed my     I would have liked to have the camera pitch up enough to show the horizon, except I didn't have enough geometry to extend the cobblestone street and the sidewalk all the way to infinity (mostly because that would require infinite stuff).  Even in that shot it is clear that I need to widen the "sidewalk" to cover up the "sky"box.    Contrast with     where the ray tracer can use the procedurally replicated geometry.   is a failed attempt at faking out the horizon, even with low detail.  It fails because lighting's interaction with the geometry gives it away.  Some techniques that prevent you from having to deal with infinity include   put the scene in a box (usually a room) put the scene in a canyon (which is just a box without a top so you can see the relatively featureless sky) route the road into a tunnel, or around a corner and hide the horizon with other geometry (which is really just a variant on the canyon). have no detail on the ground at all (flat color or gradient)   Those are all dodging the question of interest to me:  What techniques have people used in blender to fake various kinds of detail repeating into the distance (instead of just obstructing the view of infinity)? How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$ Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.   I showed it using Abel's theorem and limit comparison test. Any other simpler method? How can I evaluate a C# expression dynamically? I would like to do the equivalent of:  object result = Eval("1 + 3"); string now    = Eval("System.DateTime.Now().ToString()") as string   Following Biri s , I got this snippet (modified to remove obsolete method ICodeCompiler.CreateCompiler():  private object Eval(string sExpression) {     CSharpCodeProvider c = new CSharpCodeProvider();     CompilerParameters cp = new CompilerParameters();      cp.ReferencedAssemblies.Add("system.dll");      cp.CompilerOptions = "/t:library";     cp.GenerateInMemory = true;      StringBuilder sb = new StringBuilder("");     sb.Append("using System;\n");      sb.Append("namespace CSCodeEvaler{ \n");     sb.Append("public class CSCodeEvaler{ \n");     sb.Append("public object EvalCode(){\n");     sb.Append("return " + sExpression + "; \n");     sb.Append("} \n");     sb.Append("} \n");     sb.Append("}\n");      CompilerResults cr = c.CompileAssemblyFromSource(cp, sb.ToString());     if (cr.Errors.Count &gt; 0)     {         throw new InvalidExpressionException(             string.Format("Error ({0}) evaluating: {1}",              cr.Errors[0].ErrorText, sExpression));     }      System.Reflection.Assembly a = cr.CompiledAssembly;     object o = a.CreateInstance("CSCodeEvaler.CSCodeEvaler");      Type t = o.GetType();     MethodInfo mi = t.GetMethod("EvalCode");      object s = mi.Invoke(o, null);     return s;  } How does negation affect the use and understanding of "or" and "and" I'm trying to make more sense of how negation effects how a sentence is parsed and understood if and's and or's are used within them.  Pop quiz: You are trapped on a bus with a bomb going 50 MPH. You have a radio... (okay enough Keanu Reeves references) :-)  Let's say the bomb expert says "Don't cut the red or green wires." I think that means "Do not cut the wires colored red. Do not cut the wires colored green."  Now, let's say the bomb expert said on the radio "Do not cut the red and green wires", instead. Do I take that to mean:     "Do not cut the wires colored red. Do not cut the wires colored green."   OR     "Do not cut the wires whose insulation has red and green stripes." ?   OR     "It's okay to cut the wires colored red or the wires colored green, as long I don't cut one of each color."   Which meaning should I take if the radio breaks and I can't ask follow up questions?