There was a previous problem in my homework that basically demonstrated that:  10C7 = 9C6 + 8C6 + 7C6 + 6C6   And our question is:   "Use that fact to derive a summation formula involving expressions nC1."   I'm not entirely sure what this means, but I'm assuming we are to use Sigma.  This is what I came up with:  $${}_nC_r = \sum_{i=r - 1}^{n-1} {}_iC_{r-1}$$  I'm not sure if I'm even using legal notation here, so any help would be greatly appreciated.
After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
How do I create a master workbook in excel? In powerpoint we have something called a master slide.  I want the same idea for excel.  I want to create a template that I can work on with formulas and I want it to update all children files.  Example: I have a budget template.  This template has a budget listing all required equipments and their cost for a project.  We then have a formula that calculates the total cost of all equipment. I want to update only the template file with equipment and formulas, but I want the actual files (children) to contain the costs (because they might be different costs for different locations, but the rest of the budget and its formulas are shared).  Its pretty much exactly what a master slide in powerpoint is, if you've ever used that you should understand what I'm trying to do pretty easily.  One master file to store formulas, and a bunch of children files that store the data to be accessed by those formulas.  This way if a formula needs to be changed, the templates is adjusted and all other files update accordingly.  Any ideas?  I'd prefer not to have to use VB, and I'd also like a solution that works for both mac and windows.  Found  question but no answers.  I've already created a template, but that doesn't seem to update the other files when I modify it. Evaluation inside Button stops after some seconds What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$.  So I was wondering where I have gone wrong. Question:  Let ${a_n}$ be a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$  Prove that  $\lim_{n\to\infty}a_n^2=a^2$  Attempt: Without using any properties of limits (this is a question in a section of the book that came before any list of any limit theorems)  By definition:  $|a_n^2-a^2|&lt;\epsilon$   $|a_n^2-a^2|$ $\leq$ $|(a_n+a)(a_n-a)|$ $\leq$ $|(a_n+a)| |(a_n-a)|$ $\leq$   $(|a_n | + |a|)(|a_n -a|)$ $&lt;$ $\epsilon$  And since I know that $\lim_{n\to\infty}a_n=a$, I'll subsitute that in for $|a_n|^2$ as $a^2$   $\therefore$ $a^2-|a|^2 &lt; \epsilon$ which is true since $a^2-|a|^2$ in my case.  I wonder if this is a possible solution? Declaring Text Decorations such as Underline, Strikethrough in a Style How do I include text decorations such as Underline, Strikethrough etc in a Style definition:  &lt;Style x:Key="UnderlinedLabel"&gt;      &lt;Setter Property="Control.FontFamily" Value="Trebuchet MS" /&gt;      &lt;Setter Property="Control.FontSize" Value="14" /&gt;      &lt;!-- Next line fails --&gt;      &lt;Setter Property="Control.TextDecorations" Value="Underline" /&gt; &lt;/Style&gt;   I'm familiar with using the following XAML to underline text:  &lt;TextBlock&gt;    &lt;Underline&gt;        Underlined text    &lt;/Underline&gt; &lt;/TextBlock&gt;   However text decoration is just another style, I want to be able to define it declaritively like FontWeight, FontSize etc.  [Update]  I was applying this style to a Label control.  This was my main problem.  It appears you can't underline text in a Label.  Change to a TextBlock (thanks gix) and all is well. Fastest way to remove non-numeric characters from a VARCHAR in SQL Server I'm writing an import utility that is using phone numbers as a unique key within the import.  I need to check that the phone number does not already exist in my DB. The problem is that   phone numbers in the DB could have things like dashes and parenthesis and possibly other things. I wrote a function to remove these things, the problem is that it is slow and with thousands of records in my DB and thousands of records to import at once, this process can be unacceptably slow. I've already made the phone number column an index.  I tried using the script from this post:   But that didn't speed it up any.  Is there a faster way to remove non-numeric characters? Something that can perform well when 10,000 to 100,000 records have to be compared.  Whatever is done needs to perform fast.  Update Given what people responded with, I think I'm going to have to clean the fields before I run the import utility.   To answer the question of what I'm writing the import utility in, it is a C# app. I'm comparing BIGINT to BIGINT now, with no need to alter DB data and I'm still taking a performance hit with a very small set of data (about 2000 records).   Could comparing BIGINT to BIGINT be slowing things down?  I've optimized the code side of my app as much as I can (removed regexes, removed unneccessary DB calls). Although I can't isolate SQL as the source of the problem anymore, I still feel like it is. JavaScript editor within Eclipse So, if we have a large pipe which is constricted in the middle, the fluid in the middle section  will move faster. Why is it so? What dictates that liquid that an X amount of it must pass on the other side regardless of the size of the encasement?  The only reason I could think about is because that the initial pressure which causes the movement somehow is being transferred to a smaller area, but that makes no sense. Collisions in space/vacuum Can a PC be a lich? The heading says it all, can a PC be a lich? If so, how? The DMG hints that the Book of Vile Darkness contains the secrets of how; there is nothing official otherwise. Toggle wireframe mode with hotkey How do I toggle in and out of wireframe mode with my keyboard? Is there a hotkey for this?  I'm a beginner (obviously). I thought I could find this in the "view" menu, but it either doesn't exist or I have horrendous eyesight. Thanks! Why is oil a better lubricant than water? How come mineral oil is a better lubricant than water, even though water has a lower viscosity?    When two surfaces slide over each other with a gap filled with a fluid, the different layers of the fluid are dragged at different speeds. The very top layer touching the top metal surface will have the same speed as the surface itself, while the bottommost layer is stationary. The speed in the layers between is distributed linearly and there exist friction forces between those layers that slow the movement. Those frictional forces should be reduces however, if a fluid with a lower viscosity is chosen.   How come this is not so?   Does it have to do with water's polarity, so that it sticks to surfaces in a different way than oil? how to get a picklist all values in apex controller Difference between standard beta and unstandard beta distributions? What is the meaning of “runneth”? What is the meaning of “runneth” in ? Date Format In RUBY I've been trying to ignore my burning desire to make some guacamole because every time I go to the supermarket to buy avocados I can't manage to pick out good ones.  They always either become too soft before I have a chance to do anything with them or are hard and not very tasty.  How can I tell when an avocado is perfectly ripe? Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy  $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$  Prove that $$\det(A+B)=\det B.$$ Order of "not" with infinitive This is one thing that keeps bugging me, and maybe there's a direct answer. Grammatically, which one is more correct of these two?  Does it make a difference?  I tried not to do that. I tried to not do that. I'm looking for a way to have two separate objects be connected, so that when I move the first object the second one will move also. The term parent is used, but how do I parent a object?