WTS:  ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+...+{n-1\choose k}+{n\choose k}.$  Algebraically, it is possible, although tedious to show. Is there any combinatorial approach to show this fact?
I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out?
SOQL results returning RecordTypeId which is not included in query I am running this script from developer console.  List&lt;Account&gt; lstAccount = [SELECT Id, Name, BillingStreet From Account LIMIT 1]; System.debug(lstAccount);   Debug Log:     23:47:04:012 USER_DEBUG [5]|DEBUG|(Account:{Id=0019000001nNLGEAA4, Name=Acct_nm001, BillingStreet=xyz Road, RecordTypeId=01290000000iaNiAAI})   I never queried for RecordtypeId in SOQL, but it is included in the list.  Although, if I run the query from the query editor, it is not showing RecordtypeId column.   Is this by design? Then why not Salesforce is not providing us audit fields in the first scenario. In C++, what is a virtual base class? I want to know what a "virtual base class" is and what it means.  Let me show an example:  class Foo { public:     void DoSomething() { /* ... */ } };  class Bar : public virtual Foo { public:     void DoSpecific() { /* ... */ } }; I want wp_link_pages (mutli-page posts) to display the page numbers, the word "previous" before those numbers, and a "next" after those numbers.  It would look like this:  Prev 1, 2, 3, 4 Next  I'm attempting to do this without a plugin.  Here's what I've tried so far, but it isn't working, it is only displaying numbers.  &lt;?php wp_link_pages(array(     'before' =&gt; '&lt;span style="clear:both; display:block"&gt;Pages',      'after'  =&gt; '&lt;/span&gt;',     'next_or_number'=&gt;'number',     'previouspagelink' =&gt; 'previous',     'nextpagelink'=&gt; 'Next' )); ?&gt; AMT stock option scenario what is PGP public key block? There's PGP public key block posted on some websites (eg. ).  It's not hex code. it uses much more alphabet characters. What is it?  Why is this information posted? how can i use it? I'm aware there are been several similar questions, and some great answers with hints on how to prove this, for example  and .  But I've never really seen a detailed proof of the absolute convergence of the Riemann zeta function in the half-plane $\Re(s)&gt; 1$.  I hope there's interest on a detailed proof of this for reference. Here is my attempt. Please, fill in any detail that might be missing, or point out any mistake!  On $\Re(s)=\sigma&gt; 1$, we have  $$\sum_{n=1}^{\infty}\bigg|\frac{1}{n^s}\bigg|=\sum_{n=1}^{\infty}\frac{1}{n^\sigma}$$  and  $$\frac{1}{n^\sigma}\leq\frac{1}{n^{1+\epsilon}}$$  for any $\epsilon &gt;0$, so by the direct comparison test, if   $$\sum_{n=1}^{\infty}\frac{1}{n^{1+\epsilon}}$$  converges absolutely, then so do $\sum_{n=1}^{\infty}|1/n^s|$ and $\zeta(s)$.  By the integral test, the convergence of $\sum_{n=1}^{\infty}1/n^{1+\epsilon}$ is equivalente to the finitude of the integral  $$\int_1^\infty \frac{1}{n^{1+\epsilon}}=\bigg[-\frac{1}{\epsilon x^\epsilon}\bigg]=\frac{1}{\epsilon}$$  which holds for any $\epsilon &lt; \infty$.  Any correction or improvement is welcomed! Thanks in advance. With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons? Move Lifejacket and Lifeboat to Answer Badges section in Help Center The new blog post  mentions the new badges  and  are answer badges, but they are listed on the  under Question Badges:    Since you earn these primarily by posting an answer (you're not required to do anything about the question, though editing it will probably help), they definitely belong to the Answer Badges. Notify users of a question they closed being edited or nominated for reopening In the new canon what do we actually know about Darth Plagueis? I read and loved  but with all the recent changes to what is and isn't canon I'm losing track.  There's a common theory that Supreme Leader Snoke is Darth Plagueis, most are discounted because Plagueis is a Muun. But where does that information come from? What current canon information do we have about Plagueis that disproves this theory and what has been marked as Legends? A different approach in distributing $8$ distinct balls into $6$ distinct boxes Find the number of ways in distributing $8$ distinct balls into $6$ distinct boxes such that there is at least $1$ ball in each box.  We are well acquainted with the traditional Inclusion-Exclusion principle in solving this and it will come up as $191520$ ways.  I tried in this way:  When we distribute $8$ balls into $6$ boxes we have two possible cases  Case $1.$ There are exactly $2$ boxes with $2$ balls each  Case $2.$ There is exactly $1$ Box with $3$ balls in it.  For Case $1$, the distribution is done by choosing $2$ boxes from $6$ boxes and $2$ balls from $8$ balls and those two balls can be arranged in $2!$ ways. But I require another two balls from remaining $6$ balls to be placed in each of these two boxes which can be done in $\binom{6}{2} \times 2!$ ways. Finally remaining $4$ balls can be distributed in remaining $4$ boxes in $4!$ ways. So total number of ways for Case $1.$ is  $$\binom{6}{2} \times \binom{8}{2} \times 2! \times \binom{6}{2} \times 2! \times 4!=604800$$  For Case $2$, the number of ways is  $$\binom{6}{1} \times \binom{8}{3} \times 5!=40320$$  I am pretty sure the value obtained for Case $2.$ is correct. but I could not find where I went wrong in Case $1.$ How to construct a relative path in Java from two absolute paths (or URLs)? How do I define solar irradiance in the context of a solar cell? This is homework. I've tried my best to find the best answer that I could understand and could be accepted by my lecturer.   I've searched my lecture notes, wiki and google to find some good explanation defining solar irradiance. But somehow I still don't understand how I should 'define solar irradiance'. Can someone give me a few good sentences explaining solar irradiance. What are the changes, if any, in a photovoltaic cell (in terms of short-circuit current and open-circuit voltage) if the irradiance is reduced by 50%? Quantum computers are known to be able to crack  a broad range of cryptographic algorithms which were previously thought to be solvable only by resources increasing exponentially with the bit size of the key. An example for that is .  But, as far I know, not all problems fall into this category. On , we can read     Researchers have developed a computer algorithm that doesn’t solve problems but instead creates them for the purpose of evaluating quantum computers.   Can we still expect a new cryptographic algorithm which will be hard to crack using even a quantum computer? For clarity: the question refers to specifically to the design of new algorithms. Is "You are correct" correct? It seems to me that "You are correct" means that the person in question is supposedly "correct" in their quiddity. "That is correct" or "You are right" seems a lot better.  I also would like to know if someone is using correct(ly) correctly, as to correct someone who is potentially using correctly incorrectly. Story about workers repeatedly doing trivial work I'm trying to remember a (short) story I read in school some time ago. I don't have much, but here it is:   Numbers of workers are assigned to trivial task, like changing broken lightbulbs (or something similar). Once they finished their daily work, another group of workers are putting the broken lightbulbs in again, allowing for an endless, trivlal working circle.   I know that isn't much, but maybe someone has an idea, thanks. How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?  I ran into this identity on . I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure. Why does a minimal prime ideal consist of zerodivisors? Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.  This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent.  Hence every element of $P$ is a zero-divisor. (As Matt E has observed, when $A$ is nonnoetherian, one can still use a similar argument: $PA_P$ is the only prime in $A_P$, hence is the radical of $A_P$ by elementary commutative algebra.)  Can this be proved without using localization? $n$ choose $k$ where $n$ is less than $k$ I am working on parameter estimation and one of the estimators involves a summation of $_nC_k$ ($n$ choose $k$) expressions. For some iterations, I need to compute expressions like $_0C_1$, $_0C_2$, etc.  In general how do we compute $_nC_k$ when $n$ is less than $k$? Do we still use the formula $\frac{n!}{(n-k)!k!}$ and use the gamma function to compute the negative factorial?  Thanks! How to choose the right bean scope?