let p be a prime and let n be any integer satisfying 1&lt;= n &lt;= p-1.  Prove that p divides the binomial coefficient (p,n) = p!/[(p-n)!n!]  i know that p|p! but p does not divides 1/[(p-n)!n!] since 1/[(p-n)!n!] &lt; p.
Prove that:   $$p \,\,\left|\, {p \choose k} \right., \quad 0&lt; k \lt p$$   if $p$ is prime.   how to prove that with direct proof?
Performance differences between visibility:hidden and display:none Multiple Inheritance in C# Copy Physics To Multiple Objects Using the cell fracture addon, I created many objects from my mesh. I want to have a rigid body physics on all of them. How would I copy one of the fractured pieces physics to all the pieces? I tired ctrl L and the copy attributes addon, but none of them have the option. Any help is great. Output to stdout and at the same time grep into a file $a_1x_1+a_2x_2+a_3x_3+...+a_nx_n=$ is called a linear equation because it represents the equation of a line in an n dimensional space. So "linear" comes from the word "line".Basically there should not be any higher power of x failing which the graph of the function will not be a straight line.  simillarly  $a(x)y+b(x)y'+c(x)y"+d(x)y'''+...+q(x)=0$ is also called linear differential equation because all the derivatives have power equal to 1 which is similar to the above definition of a linear equation.  A function f is called linear if: $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$. Here c is a constant. In this definition of linearity of function "$f$" what does the word linear means? How does it relate to a straight line?  Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?  So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line? At Least please explain how the linearity of function f and linear vector space relate to the equation of a line. Card example:   There are a lot of cards that use counter as a noun. I did some Googling but I'm just not clear on what I need to use for counters.  Are they physical things I can buy and if so, what is a good online resource?  Can I just make do with coins or something? How to turn cheats on or off Does a man contain all the genes needed to make a woman? This question is brought on by a Sci Fi novel I am thinking about writing. The plot device involves a colonist in charge of building a population on a new planet who loses his supply of embryos and so forth in the landing. With the artificial "wombs" intact, he populates the planet with clones of himself.  If he wanted to introduce enough variation (without making up new genes, only rearranging) in the population to allow them to be able to reproduce naturally without inbreeding problems, would he be able to? In particular, could he make a woman?  Alternately, could a woman (without a Y chromosome) do the the inverse?  At the risk of introducing bias into the answers, the plot as it stands assumes he is a male and he CAN'T do any of the above and he is restricted to clones. Using the properties panel I can add, with "+"  button, only eight UV maps. Is it possible to have more than 8 of them? Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent There's a new popup helping you with searching, but one of the hints is wrong:    Searching for  will return answers as well as questions, and it's not the number of (up/down)votes which counts, it's the score. Better would be "posts with a score of 3 or higher" or "questions and answers with a score of 3 or higher". Is propagation of uncertainties linear? "Fake" 3d effect in Photoshop/Illustrator with raster images Please provide me with the guidance to install oracle 11gr2 on Ubuntu 14.04. Is password confirmation necessary on a Change Password dialog? On one hand there is no need to ask for password confirmation because it is annoying. If I made a mistake when confirming my password I'll need retype my very complex password again. There is also a button that can show/hide the password so I don't need retype it in order to confirm it.   However, some users don't use the password show/hide button and the 'forgotten password' option will be pain for them.  Twitter removed the password confirmation from registration but left the dual confirmation in place for changing password. Is this right approach? Windows Backup & Restore: What to Keep from AppData? A couple years ago, my father made a backup of one of our computers using Windows Backup and Restore. It was either Windows Vista or Windows 7, not sure which. He backed up the entire computer, including Program Files, AppData, My Documents, everything, and now it is on .zip files on a WD Passport with 300GB, I'm not sure exactly what it is called. Now I am trying to decide which files to keep and which to delete. Most of what we would keep are files, as we would need CD's to use the programs, and I'm pretty sure they would install everything all over again. That pretty much leaves out Program Files, however, I'm not sure about AppData. For example, my mother used Greetings Workshop and wants some files from there, and she may want to keep some default settings she used. I am talking generally here, as I'm sure some programs have different ways of saving things in AppData.  What, if anything, should I keep from AppData, and what should I delete? I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point? Why does the air flow faster over the top of an airfoil? I understand the common explanation of lift, which describes the airflow over the top of the wing as moving faster than the air below the wing.  However, I don't quite understand why the air moves faster.  I've read some explanations citing a circulation effect among others, but haven't found a good, clear explanation for the exact cause.  Can someone help me out? I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$?  Thanks for anyone who give me references for a 'simple proof' of these results Why did Owen Chase not fire his harpoon at Moby Dick? In movie In the Heart of the Sea, when Owen Chase finally faces the big white whale (later named Moby Dick) with his harpoon, why does he not do it?