Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
Prove universal morphism is unique up to unique isomorphism. Combining Shimano RS500 crankset with Stages Shimano 105 5800 I've been thinking about getting a power meter for a while now. I finally decided I would get one, but now I realise that Stages only starts at the Shimano 105 level.  I currently have a Shimano RS500 Crankset ( 50 x 34 ) and I'm very pleased with it, so I don't feel a need to replace the whole crankset with a 105 5800.  I also have no plans to upgrade anytime soon to Dura Ace or something like that (the rest of my bike has the 105 5800 set), so I don't think I should buy a higher lever crank arm from Stages.  Now, I know something about bikes, but not a lot. It seems that what I have (RS500) looks compatible with the 105 crank arm (just judging by the looks), but I would like to be certain before I spend this amount of money.  My set:  Stages I would like to get:   Kind regards and thanks in advance, So I am running on Windows 7 (64-bit) Home Premium with 12 GB RAM (Intel Core i7 920).  I have restricted the Page File to 400MB since it was eating up a lot of space om my SSD (that is 80GB).   After that I sometimes get the "low memory"-warning, like this, except for the behold-comment:   When I get those warning I have usually a lot of RAM available; when I check Resource Manager I have over 2GB "Free" and over 2GB "Available" - usually more than that.  The diskspace on all my drives have over 10GB free.  So the question is - why does Windows complain? It actually restarted by computer (hard boot) when it happened, and as I said - lots or RAM available. We have in and out degree of a directed graph G. if G does not includes loop (edge from one vertex to itself) and does not include multiple edge (from each vertex to another vertex at most one directed edge), we want to check for how many of the following we have a corresponding graph. the vertex number start from 1 to n and the degree sequence are sort by vertex numbers.  a) $d_{in}=(0,1,2,3), d_{out}=(2,2,1,1)$  b) $d_{in}=(2,2,1), d_{out}=(2,2,1)$  c) $d_{in}=(1,1,2,3,3), d_{out}=(2,2,3, 1,2)$  I want to find a nice way instead of drawing graph.  for (C): Number of total possibilities for an equation I need to find the number of possibilities for which the following equation exists:  $$x_1 + x_2 + x_3 + \cdots + x_{10} \leq 70$$  Each variable is a non-negative integer.  I tried simplifying the question to the point of finding the number of possible solutions to each equation seperatly:  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 70,$$ which is ${79}\choose{10}$.  $$x_1 + x_2 + x_3 \cdots + x_{10} = 69,$$ which is ${78}\choose{10}$.  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 68,$$ which is ${77}\choose{10}$, and so forth.  Then I thought about adding all of these together. This seems a bit too much work considering that the textbook solution is: ${80}\choose{10}$, and I can't seem to figure out the idiomatic way of approaching these kind of questions.   Any help is greatly appreciated. For thematic reasons, I am wanting to make a Zatoichi-style character based on using  and Decisive Strike monk alternate feature from the PHB2. The only catch is that Decisive Strike says:     . . . make one attack with an unarmed strike or a special monk weapon . . .   What are all of the official ways to add a weapon to the list of ? I have found only one in the Shou Disciple prestige class (Unapproachable East, p.32), but doing so would mean the earliest the character could use decisive strike with a katana would be level 10. I would like to be doing it no later than level 5 if such an option exists.  Note that, for the purposes of this question, anything that works for "Flurry of Blows" can be interchanged with "Decisive Strike" (e.g. the "Martial Flurry" ability of a Shou Disciple gains ability to use other weapons with "Flurry of Blows," but that is fine). Partition of a set and Hall's theorem I have recently a purchased Lenovo Thinkpad E480 laptop, I have installed Ubuntu 18.04 on it, everything is working fine except am not able to connect to network via WiFi. Please help.  I followed this , but am unable to solve the issue.  Below are the outputs of some commands.  lsb_release -a  No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 18.04.1 LTS Release:    18.04 Codename:   bionic   uname -r  4.15.0-33-generic   rfkill list all  0: tpacpi_bluetooth_sw: Bluetooth     Soft blocked: no     Hard blocked: no 1: hci0: Bluetooth     Soft blocked: no     Hard blocked: no   lspci -nnk | grep 0280 -A3  05:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821CE 802.11ac PCIe Wireless Network Adapter [10ec:c821]     Subsystem: Lenovo RTL8821CE 802.11ac PCIe Wireless Network Adapter [17aa:c024] 06:00.0 SD Host controller [0805]: O2 Micro, Inc. SD/MMC Card Reader Controller [1217:8621] (rev 01)     Subsystem: Lenovo SD/MMC Card Reader Controller [17aa:5068]   sudo lshw -C network   *-network                         description: Ethernet interface        product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: enp3s0        version: 10        serial: 8c:16:45:a3:d9:f1        size: 100Mbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168g-3_0.0.1 04/23/13 ip=192.168.2.2 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s        resources: irq:16 ioport:d000(size=256) memory:f1204000-f1204fff memory:f1200000-f1203fff   *-network UNCLAIMED        description: Network controller        product: Realtek Semiconductor Co., Ltd.        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:05:00.0        version: 00        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: ioport:c000(size=256) memory:f1100000-f110ffff   It seems I have to clone, compile and install the required WiFi driver, please guide me which WiFi driver to use and how to install (with steps if possible).  P.S.: Let me know if any further information is needed to solve this issue, I will add the required info (if any) in the question as per the guidance given. Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you. I understand that the vibration modes of an elementary string determines the identity of its particle. When I first heard this, I visualized a stiff circular steel band-like entity vibrating with 2, 3, 4, 5,... nodes or antinodes, the number determining the particle. This was further supported when I asked a well-known physicist how one gets from equations to physical strings, and he said he recognized the harmonic nature of the solutions.  Bu then I noticed that physicists seem to accept the image of a soft spaghetti-like string flopping about with no well-defined vibrational modes. I asked another physicist about this, and his answer was simply that a string has multiple modes, not necessarily in a harmonic progression.  So my questions are 1) How do you visualize the vibrations of a string, and 2) how are they related to particle identity? I don't understand this and I'm now worried as I have donated to a lot of servers and I don't want to lose this account. On the launcher it says my new username but when I log on it says "Account not Authenticated with minecraft.net" and "Non-Valid Username".  How can I regain the ability to use my account online? I have a background in QFT, GR and differential geometry at the level of a master student in theoretical physics. I would like to touch the area of CFT. I know the textbook of Philippe Di Francesco. It may be too big for the beginner like me. Are there some good introductory lectures or textbooks adapted to the needs of beginners? Create a tar archive split into blocks of a maximum size I need to backup a fairly large directory, but I am limited by the size of individual files. I'd like to essentially create a tar.(gz|bz2) archive which is split into 200MB maximum archives. Clonezilla does something similar to this by splitting image backups named like so:  sda1.backup.tar.gz.aa sda1.backup.tar.gz.ab sda1.backup.tar.gz.ac   Is there a way I can do this in one command? I understand how to use the split command, but I'd like to not have to create one giant archive, then split it into smaller archives, as this would double the disk space I'd need in order to initially create the archive. In some cases, questions and answers that have been deleted are still showing up on /review pages. They do not have "Improve" buttons, but they can still be approved or rejected. I'm not sure what causes a post to end up in this limbo state instead of being removed from the queue entirely, which I think is what's supposed to happen.  The suggested edit I noticed this on was , but it just leads to a "Page Not Found" message now.  Tim Stone provided the following screenshot of another instance : Is the following just a matter of showing the 3 properties that make up a metric??  Define d on $\Bbb R\times\Bbb R$ by $d(x,y)=\min \{1,|x-y|\}$. Show that $d$ is a metric on $\Bbb R$     $d(x,y)=0$ if $x=y$   $d(x,y)=d(y,x)$ for every $x,y \in X$   $d(x,y)\le d(x,z)+d(z,y)$ for every $x,y,z \in X$ Read two column CSV as dict with 1st column as key Which predictions of The Last Jedi: Darths & Droids Bingo were correct? Calculate text width with JavaScript How do I specify a pointer to an overloaded function? How to peel hard boiled eggs easily? This is one of the most annoying duties in the kitchen, for me.  What are yours tips and tricks to peel hard boiled eggs easily?