I have to prove that: $$ \binom n 0 ^2  + \binom n 1 ^2  + \dots + \binom n n ^2 = \binom { 2n} n $$  I don't want a complete solution, but only a hint.
I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$.
Why do carnivores evolve? If humans can live without consuming other animals, then why do we do it? From a biological point of view, why do we eat meat? I would also extend the question to other animals because many animals are carnivorous: why did carnivory evolve and why do animals eat meat? If $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$   then prove that $f(x)=0$ for all $x \in [a,b]$.  I tried with Riemann's definite integral definition but couldn't proceed Is there a way to format a UTC time into any arbitrary string format I want in java?  Basically I was thinking of having some class take the timestamp and I pass it is string telling it how I want it formated, and it returns the formatted string for me.  Is there a way to do this? What is the highest power of n in $(n^r-1)!$ where $n$, $r$ are positive integers? The answer supplied is $\frac{n^r-nr+r-1}{n-1}$   The highest power of $n$ in $n^r!  = r +$ the highest power of $n$ in $(n^r-1)!$ as $n^r$ contributes $r$ powers.    So,  the highest power of $n$ in $n^r!$ needs to be $\frac{n^r-nr+r-1}{n-1} + r =\sum_{1≤t&lt;r}n^t $.    This is clearly true if $n$ were prime according to .  Again,  if $r=1$, $\frac{n^r-nr+r-1}{n-1}$ becomes $0$.    But according to , the  highest power of composite $n$ in $(n-1)! ≥ 1 $.  So, the answer does not seem to hold for $r=1$ if $n$ is composite. If $n=\prod p_i^{a_i}$, where $p_i$ are distinct primes,  $n^r=\prod p_i^{ra_i}$  So, the highest power $(q_i)$ of $p_i$ in  $n^r!$ is  $$\sum_{1≤t&lt;∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor =\sum_{1≤t≤ra_i}\frac{n^r}{p_i^t} +\sum_{ra_i+1≤t&lt;∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor$$  as $p^t|n^r$ for $t≤ra_i$.  So, the highest power of $p_i^{a_i}$ in  $n^r!$ is $\left\lfloor \frac{q_i}{a_i} \right\rfloor=P_i$(say).   So, the highest power of n=$\prod p_i^{a_i}$ in  $n^r!$ is $min(P_i)$  I could not proceed anymore. Is there a mass spectrometer software that can measure the mass of a proton Is there a way to recover files from a storage device partially overwritten with dd? I'm an idiot, I was writing an image to a USB drive using dd when instead of typing /dev/sdb I typed /dev/sdd and wrote the 135mb image to my 750gb External Hardrive full of files. Is there ANY possibility of recovering the files or did dd completely zero out the drive? Please help, as this is one of the lowest points of my life EVER.  dd finished writing the 135mb image in 5 seconds, so I didn't have a chance to stop it. By the time I knew what happened to was too late. The command was as follows:   sudo dd if=generic.img of=/dev/sdd bs=1M   EDIT2: The External Harddrive's filesystem was FAT32(I've had the drive for awhile and FAT32 was the most Universal FS at the time). I also have a 2TB drive I just got which I was planning on moving most of the files to from the 750gb, If needed I can use it to transfer the Recovered files to. I had installed VirtualBox but it started giving me problems. So I uninstalled it using:-  sudo apt-get remove virtualbox-\* sudo apt-get purge virtualbox-\*   But when I reinstalled VirtualBox it again resumed to previous settings and started giving same problems. How can i completely remove VirtualBox from my laptop and then reinstall it such that it wont give same problem again? What software do you use to keep up to date on literature, and if Google Scholar, how do you avoid information overload? How to upgrade only core with drush? Why the statement "there exist at least one bound state for negative/attractive potential" doesn't hold for 3D case? Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal.   However, today I'm doing a homework considering a potential like this:$$V(r)=-V_0\quad(r&lt;a)$$$$ V(r)=0\quad(r&gt;a)$$ and found that there is no bound state when $V_0a^2&lt;\pi^2\hbar^2/8m$.  So what's the condition that we have at least one bound state for 3D and 2D? We all know that Superman changes in the phone booth. But where does he put his regular clothes while he's on a mission? He doesn't throw it out, because he changes back. But we don't see him carrying a bag of dirty laundry while he's on a mission.  ("In his right hand, Superman has Lois Lane, and in his left hand he seems to be holding... boxers??)  And I can't imagine he leaves it in the phone booth.  So where does Superman leave his regular clothes after he changes? I have  a  requirement where a future method need to be called from batch apex ,I came across various articles stating that a future method could not be called from the batch. Is there any alternate solutions for that? OS volume shows as type 'FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF' Yesterday, I have tried to delete an Ubuntu partition. Since, my Mac (13", MBPro mid-2014) doesn't boot. Now, I boot on Mac OS copy with an USB. i follow different subject of this forum (like this : ), and my volume partition change from FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF to 48465300-0000-11AA-AA11-00306543ECAC. But I still can't boot to it...    Thank you very much ! How would I remove the border from an iframe embedded in my web app? An example of the iframe is:  &lt;iframe src="myURL" width="300" height="300"&gt;Browser not compatible.&lt;/iframe&gt;   I would like the transition from the content on my page to the contents of the iframe to be seamless, assuming the background colors are consistent. The target browser is IE6 only and unfortunately solutions for others will not help. Finding net magnetic and electric force on charged particle This is from my textbook, it is not an assigned problem, but I want to understand.   It says:      Consider the situation in the figure, in which there is a uniform electric field in the x direction and a uniform magnetic field in the y direction. For each example of a proton at rest or moving in the x, y, or z direction, what is the direction of the net electric and magnetic force on the proton at this instant?   I believe I need to use the equation   $$ F_{net}=(q\overrightarrow{E})+q(\overrightarrow{v} \times \overrightarrow{B}) $$  But I'm not sure exactly how. We've just started to learn about this, and I want to get a head start. Could anyone put me on the right track? Just want to get clarification on something we're noticing in our org. Its regarding a new integration user we've created for an external application to connect to Salesforce. We also created a Connected App for this application to authenticate using OAuth. All of this is setup and working up to this point.   But one thing that we don't understand is there is a related list on the integration user detail page named "OAuth Connected Apps". We're concerned because although we are connecting fine and data is getting return to the external application no records are showing up in this "OAuth Connected Apps" related list. Should we expect to see Authorization or Access tokens in this list?   We just want to make sure we aren't bypassing the connected app in anyway, which we assume not, but just want to run it by the community and see. All comments are appreciated. Thanks ahead of time for the support. In GLM regression I have always been told to set the reference level of categorical/ordinal/dummy variables to the level with the most exposure (level with most data), because this somehow makes the model more stable. Is there any statistical reason behind this, or just historical practice without statistical reason?  I can understand that one wants a "baselevel" (reference level) in the model that are most likely to be correct, but if you are not sure about the other levels, then they surely should not be included in the regression in the first place? If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in all of $L_i$. Does this also hold if the $L_i$ are not Galois extensions of $K$?  The proof that I know regarding the compositum uses the fact that the $L_i/K$ are Galois, so is this true in the more general setting and how would one prove it? Can I use my single entry Schengen visa to go to a different Schengen country than originally planned? My friends and I planned to visit Germany, so I got a single entry tourist Schengen visa to Germany. However, my travel plans have changed since then, because my friends are going to Italy instead. I don't have time to apply for a visa to Italy, but I have been told that I can use my Schengen visa to enter Italy. Is that true? Elevating process privilege programmatically?