I'm working on a question below:     Let $H(n,k)$ denote the number of ways to partition a set with $n$ elements into $k$ subsets of the same size. Derive a formula for $H(n,k)$.   Thanks in advance for any hints! :D
Let's have 2 numbers, N and K, where K divides N. The number of K-combinations from a given set S of N elements is a well known formula. Let's concatenate N/K groups (resulting in N elements) such as the resulting set is N. How many possibilities are there, i.e. what's the formula?  For instance:    N=4, K=2, C(4, 2) = 6 {1,2},{1,3},{1,4},{2,3},{2,4},{3,4}   The 3 possibilities are:     {1,2},{3,4}   {1,3},{2,4}   {1,4},{2,3}   I generated these combinations and I think the number goes like this:  (4, 2): 3 (6, 3): 10 (6, 2): 15 (10, 5): 126 (9, 3): 280 (10, 2): 945 (14, 7): 1716 (12, 4): 5775 (15, 5): 126126 (15, 3): 1401400   Apparently, the result always divides with (N-1).
Show that $R/I\otimes_R R/J\cong R/(I+J)$ Can you use another material as input to material nodes in cycles? How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon?   I proved geometrically that $\sin x&lt;x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin x_2|&lt;|x_1|+|x_2|$$   But this doesn't help me much finding a delta...   Thanks for any help!  P.S. I'm only at the beginning of calculus so I can't use many theorems and derivation  (because they haven't been regorously proven). If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example. This is a proof question and I am not sure how to prove it. It is obviously true if you start with $A = 0$ and square it.   I was thinking:  If $ A^2 = 0 $  then $ A A = 0 $  $ A A A^{-1} = 0 A^{-1}$  $I\,A = 0 $   but the zero matrix is not invertible and that it was not among the given conditions.  Where's a good place to start? How to kill The Twins In Terraria In Terraria I often try to defeat spazmatism first, but when he transformed, his flamethrower killed me. What is easiest way to kill The Twins? I have a long sequence of hex digits in a string, such as      000000000000484240FA063DE5D0B744ADBED63A81FAEA390000C8428640A43D5005BD44   only much longer, several kilobytes.  Is there a builtin way to convert this to a bytes object in python 2.6/3? Can others in my party go through TSA PreCheck? If I have  (i.e. I have a KTN from TSA and the "pre-check" logo gets printed on my boarding pass) and a travel companion does not, can they go with me through the PreCheck security lines? This is assuming that the travel companion is booked in the same reservation on the same flights and is also a U.S. citizen and not on any no-fly or otherwise increased scrutiny lists.  I'm guessing the answer is no, but I haven't seen anything from TSA specifically addressing this. Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that $E=\{x\in \mathbb{R}\mid f(x)=0\}$.   Since $E^c$ is open in $\mathbb{R}$, $E^c$ is a countable disjoint union of open intervals $(a_i,b_i)$. On an interval $(a,b)$, if we let $f(x)=\exp(\frac{1}{(x-a)(x-b)})$, $x\in(a,b)$; $f(x)=0$, $x\notin(a,b)$, ($f(x)=\exp(\frac{1}{(x-a)})$ or $f(x)=\exp(\frac{1}{(x-b)})$ on $(a,\infty)$ and $(-\infty,b)$ resp.), then $f$ is smooth. Hence we can get a smooth function on $\mathbb{R}$ such that $E$ is the zero point sets of $f$.  (1). How about a closed subset $E$ of $\mathbb{R}^n$? is it true that for any closed subset $E\subseteq \mathbb{R}^n$, there exists a smooth function $f$ on $\mathbb{R}^n$ such that $E=\{x\in \mathbb{R}^n\mid f(x)=0\}$?  An open set in $\mathbb{R}^n$ cannot be written as a disjoint union of countable open balls hence the proof above is not valid. The partition of unity only claims that for an  open set $U$, there exists open $V$ in $U$ such that $\bar V\subseteq U$ and a smooth function $f$ on $\mathbb{R}^n$ such that $supp f\subseteq U$, $f|_V=1$.   (2). The taylor series of $f(x)=\exp(\frac{1}{(x-a)(x-b)})$ does not converge around points $a,b$ hence $f$ is not analytic at $a,b$. Is it true that for any closed subset $E\subseteq \mathbb{R}$, there exists an analytic function $f$ on $\mathbb{R}$ such that $E=\{x\in \mathbb{R}\mid f(x)=0\}$? Cartoon with a young unicorn and a demon In the late 80s or early 90s I remember watching an animated series that followed a young unicorn and a demon.   In one episode the unicorn gave the demon his horn, and the unicorn almost died.   In another episode there was a man who was able to turn people into human shaped building blocks and built a huge tower from them.   Those are the two scenes I remember with any clarity.   Does anyone have any idea? When to use Path in WPF Binding? How do the Planets and Sun get their initial rotation? Why do Venus and Mercury rotate so slowly compared to other planets?  Why does Venus rotate in a different direction to Mercury, Earth and Mars? What will reduce my rendertime? What would reduce the render time? Should I get a better GPU or more RAM? How is the minimum of a set of IID random variables distributed? If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general? My code runs inside a JAR file, say foo.jar, and I need to know, in the code, in which folder the running foo.jar is.  So, if foo.jar is in C:\FOO\, I want to get that path no matter what my current working directory is. How do I detect when someone shakes an iPhone? I want to react when somebody shakes the iPhone. I don't particularly care how they shake it, just that it was waved vigorously about for a split second. Does anyone know how to detect this? When two accounts are merged, all the posts/votes are moved over to the "master" account, and the other account ("slave"?) is deleted. The merge is propagated network-wide, so a similar thing happens on each site.  But, this does not happen to the chat user. The "slave" chat user is kept, with a 404 parent profile link. I feel that the chat user should be merged as well, re attributing all the chat messages.  Thoughts? Can "what kind" be plural? Is it correct to say, "What kind of patents are being issued in these sophisticated times?" It seems like it should be What kind of patents is being issued..., but that sounds wrong.  What kind of patent is being issued... is correct and sounds right, but loses a little bit of the original meaning.  Is there something magic about what kind that allows it to be plural, or do people just misuse it a lot? List of Ubuntu Versions with Corresponding Linux Kernel Version Random real numbers that sum up to specific value "An hour" or "a hour"