I was given this formular, and am now tasked to use double counting to prove it:  $\sum_{k=1}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$  Quite frankly, I have no clue on how to work this out. Of course, you could start out by looking at the numbers this produces:  n     result 1     1 2     4 3     12 4     32 5     80 6     192 7     448 8     1024 9     2304 10    5120   As far as I've understood double counting, there's two ways you can use it.   The first way is to imagine a matrix where you either put a tick into a field, or you leave it empty. Then counting the ticks in the rows and the columns individually must lead to the same result. Unfortunately, I see no way to apply this concept to this particular task.  The second way is to look for an intuitional problem that can be solved by either of the two ways shown in the equation. Again, I fail to come up with anything that makes sense; especially with the fact, that the sum in the left term makes everything even more complicated.  Do you have any ideas on how to solve this problem?
Can someone explain how combinatorial proofs work?  I've included an example questions that's been giving me a hard time.  Any insight on the topic would be great.  $$\sum_{k=1}^{n}k{n \choose k} = n2^{n-1}$$  Another one:  $${2n \choose 2} = 2{n \choose 2} + n^2$$  The solution says:  LHS: Pick two people from a pool of n men and n women. RHS: Pick two women, pick two men, and pick one of each.  But I don't see how this applies.  How does $n^2$ show that we're picking "one of each?"  Thanks for the help :)
Change Windows 7/8/10 system language - welcome screen, login, etc How to assign vertex groups and bone weights through Blender's Python API? I'm making a script to import custom objects, it is still at early stage but so far vertices, faces and uv are created correctly. I want to do weight painting through Blender's Python API but I don't know how to do it properly.  Below an extract of the code that create the objects, how I could add vertex group and relative weights? I know how to extract vertex indices and weights from the input model but don't know how to use them,   (Blender version is 2.72)      for i in range(0,self. subsetCount):         subset = self.subset[i]         coords = subset.getV3(0)         faces = subset.getIndices()         me = bpy.data.meshes.new(subset.name+"_Mesh")          ob = bpy.data.objects.new(str(i).zfill(2)+'_'+subset.name, me)         ob.location = (0, 0, 0)         bpy.context.scene.objects.link(ob)          me.from_pydata(coords, [], faces)         me.update(calc_edges=True)          mat = bpy.data.materials.new(subset.name)         mat.diffuse_color = (random.uniform(0.0, 1.0), \                              random.uniform(0.0, 1.0), \                              random.uniform(0.0, 1.0))         me.materials.append(mat)          for f in me.polygons:             f.material_index = 0             f.use_smooth = 1          # vert_uvs = subset.getV2(0x28)         # if len(vert_uvs) &gt; 0:         #     me.uv_textures.new("UV0")         #     uvlist = [uv for pair in [vert_uvs[l.vertex_index] for l in me.loops] for uv in pair]         #     me.uv_layers[-1].data.foreach_set("uv", uvlist)   edit  found this to add the group:          vxgroup = ob.vertex_groups.new("group")   still don't know how to set weights for vertices (best would be at once)  edit  I was hoping to use something like a foreach_set but can't figure out how to use it, instead solved this way:          for grp in ob.vertex_groups:             grp.add(range(0,len(me.vertices)), 1.0, 'REPLACE')   is just an example and set all vertices weight in all groups of the object ob (if you try it in console select another mesh then back to see the weights applied). A symmetry of graph $X$ is a permutation of the vertices that also happens to be a permutation of the induced edges. In particular, the distances between vertices are preserved by a symmetry. Show that the set of symmetries of $X$ is a permutation group of $\operatorname{V}(X)$. Compute the cycle index of the group for the Petersen graph.   Alright I am super confused. I have no idea how to even start. Can someone point me in the right direction. How do I show that something is a permutation group? How do I compute the cycle index of a graph? Let's say I have some XML like this  &lt;channel&gt;   &lt;item&gt;     &lt;title&gt;This is title 1&lt;/title&gt;   &lt;/item&gt; &lt;/channel&gt;   The code below does what I want in that it outputs the title as a string  $xml = simplexml_load_string($xmlstring); echo $xml-&gt;channel-&gt;item-&gt;title;   Here's my problem.  The code below doesn't treat the title as a string in that context so I end up with a SimpleXML object in the array instead of a string.  $foo = array( $xml-&gt;channel-&gt;item-&gt;title );   I've been working around it like this  $foo = array( sprintf("%s",$xml-&gt;channel-&gt;item-&gt;title) );   but that seems ugly.  What's the best way to force a SimpleXML object to a string, regardless of context? Possible Duplicate:          There are a lot of really screwed up configurations out there, because people didn't take a minute to sit down and think about something as simple as the name for their Active Directory forest. I'm asking this question to self-answer so that this knowledge can be more readily available. If you think you have a better answer, by all means add your input (but be prepared to defend it!)   So, what should I name my new Active Directory? The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering map.  I am just wondering is there an easy counter example to show the necessary of "inverse image of every point is finite" condition?  I have already checked Hatcher's book (the wedged circle example) but I wonder is there a less complex and more obvious example?  I tried to construct a counter example by composition of a covering map from $R_+$ to $S^1$ with $R$ in the middle, but I failed due to I can't find a covering map from $R^+$ to $R$ which does the job. Q&A Identification of seldom man-made physical objects not being piece of art? Is SE Q&amp;A format relevant for "identification of seldom physical objects observed in real world"?  Questions like "I do not know function of this object (picture) - which purpose does it have?"  In many cases I suppose these question could have civil engineering or handcraft i.e. cultural/historical background. In second case "SE history" might be okay. For first case, I do not know. Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? I'm considering finite groups only. Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$. I think it's interesting to ask for smaller values of $n$ for which $G$ is a subgroup of $S_n$. Obviously, it's not always possible to do better than Cayley's theorem. But sometimes it is possible (for example, $\mathbb{Z}_6$ as a subgroup of $S_5$).  So I'm asking:        Given a finite group $G$, is there an algorithmic way to find or approximate the minimal $n$ for which $G$ is isomorphic to a subgroup of $S_n$?   If the answer to $(1)$ is not known, is it known for specific classes of groups?   In particular, for finite abelian groups, is it true that for a prime $p$, the minimal $n$ for $\mathbb{Z}_{p^{t_1}} \times \mathbb{Z}_{p^{t_2}}$ is $p^{t_1}+p^{t_2}$ (I can prove that is is true for different primes $p_1$ and $p_2$, but have problems when it's the same prime in both factors).      Thanks! GTA 5 Social club I used to play gta 5 online on my ps3 but it ended up breaking and I can't use it anymore I purchased the game for xbox 360 and started playing the game all over again and some of you know how hard that can be specially online.. So I was wondering if I could somehow transfer my old ps3 social club account to my xbox an use it on xbox from now on? I asked  that turned out to have multiple correct answers. How do I pick between them to determine which to accept?  All of the answers were different. They were not the same answer given multiple times. What time does sound good to you?     What time sounds good to you?    Is there a difference? What happened to Jarvis at the end of Avengers: Age of Ultron Let $f(\sin x)$ be a given function of $\sin x$.   How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$? Is the empty set a group? How do you become friends with someone? How to generate a random snowflake database disk image is malformed with dnf on Fedora 29 How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don't exactly know how that applies. Thanks for any input. What are the mechanics of time dilation and length contraction? Going beyond the mathematical equations involving light and the "speed limit of the universe", what is observed is merely a phenomenon and not a true explanation of why time dilates or length contracts.  It has been proven to work out, but do we know why?  Is it something that happens at a subatomic level?