I am having an issue with understanding how to calculate a specific case of the Coupon Collector's Problem. Say I have a set of 198 coupons. I learned how to find the estimated amount of draws to see all 198 coupons, using the following equation:  $$n \sum_{k=1}^n\frac1k$$  It turns out that for $n = 198$, the expected number of draws is approximately 1162. Let's assume, however, that I already have some of the coupons, say 50. How should I go about solving the same problem, given that I've already collected $X$ of them?
What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic?
Blog Overflow database error , which is linked from site-specific blogs, says  Error establishing a database connection  I believe this is an error. In Minecraft, is there any way to make tools indestructible (without using mods)? I want to make some tools indestructible (for an adventure map). Is it possible to do this using enchantments (or a command of some kind?) I don't know if this is a famous problem but my professor assigned this for a research project and I am hitting a dead end( I don't know what to do next).   Note: In $\phi(n)$ ; $n=rows=columns$ and this function throws out total combination of matrices that satisfies the following rule:  Rule for this square binary matrix:  Every Row and Column has two 1's.  So,    $$\phi(2)=1$$ \begin{bmatrix}    1 1 \\    1 1 \\ \end{bmatrix} $$\phi(3)=6$$  I can't write out all the possible matrices for $n=3$ but this is one of them (btw i did this by hand): \begin{bmatrix}    0 1 1 \\    1 0 1 \\ 1 1 0 \\ \end{bmatrix}   According to a program made by my friend, he told me:   $$\phi(4)=90$$ $$\phi(5)=2040$$ Moving status bar in iOS 7 $g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$ I would like to know how to unload an assembly that is loaded into the main AppDomain.  I have the following code:  var assembly = Assembly.LoadFrom( FilePathHere );   I need/want to be able to unload this assembly when I am done.  Thanks for your help. Dark color scheme for Eclipse Is Eclipse at all theme-able? I would like to install a dark color scheme for it, since I much prefer white text on dark background than the other way around. How to get my own IP address and save it to a variable in a shell script? JSON URL from StackExchange API returning jibberish? Why does C/C++ differentiates in case of array index out of bound #include &lt;stdio.h&gt; int main() {     int a[10];     a[3]=4;     a[11]=3;//does not give segmentation fault     a[25]=4;//does not give segmentation fault     a[20000]=3; //gives segmentation fault     return 0; }  I understand that it's trying to access memory allocated to process or thread in case of a[11] or a[25] and it's going out of stack bounds in case of a[20000]. Why doesn't compiler or linker give an error, aren't they aware of the array size? If not then how does sizeof(a) work correctly? Alias and functions From the bash manual     The rules concerning the definition and use of aliases are somewhat   confusing. Bash always reads at least one complete line of input   before executing any of the commands on that line. Aliases are   expanded when a command is read, not when it is executed. Therefore,   an alias definition appearing on the same line as another command does   not take effect until the next line of input is read. The commands   following the alias definition on that line are not affected by the   new alias. This behavior is also an issue when functions are executed.   Aliases are expanded when a function definition is read, not when the function is executed, because a function definition is itself a   compound command. As a consequence,   aliases defined in a function are not available until after that function is executed. To be safe, always put alias definitions on a   separate line, and do not use alias in compound commands.   The two sentences "Aliases are expanded when a function definition is read, not when the function is executed" and "aliases defined in a function are not available until after that function is executed" seem to be contrary to each other.  Can you explain what they mean respectively? New messages flagged blue bubble does not go away Exists a dedicated non-O(n²) Algorithm for avoiding to have overlapping elements? I am generating positions in a random fashion.  In the game world, instantiated elements have a radius, and shouldn't overlap.  The problem is that they will overlap because I use a usual random number generator combined with other fanciness. I can't use other random number generation algorithms where I can guarantee that the elements will never overlap.  A way to solve this is to use a spatial data structure and search for the nearest neighbors in that tree. When I insert an element into that tree, I update it accordingly.  This solution would do the work and it would be fast but my question is:  Is there a dedicated algorithm without that Tree implementation complexity which is faster than O(n²)? How to roll perception checks for characters who aren't actively looking without arousing suspicion? This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge This is a homework problem for an undergrad topology course.  Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in l^{2}$.  Claim: $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges.  I've done a couple of examples where $a_{n}$ and $b_{n}$ are general harmonic series. I've also written out the definition of convergence for $a_{n}$ and $b_{n} $. My problem is that thus far I do not understand on any level why my claim is true. Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. When initiating a battle it starts off in AR mode, but then warns me saying my phone's orientation is not detected. I am then forced to turn off AR mode. Has anybody experienced this? If so, what are the solutions?    Per comments, I am unsure this is a permissions issue. I have everything turned on. Is there a way to render an object with a subdivision surface modifier with a wireframe which is the original geometry as well as the wires that make the subdivision surface?  Example:   This wireframe here is curved. What if there was displayed the straight wireframe. The wires that make are made by the subdivision surface. Thats what I want to achieve. How to do it?  What I mean is - How to render the curved wireframe (which is the original geometry with the straight wireframe (which is the wireframe the is made by the subdivision surface)? Make a momentary switch control a toggle What are the simplest, cheapest, smallest ways to make a momentary switch produce a 2-state toggling output (latching momentary switch)?  In other words, the output is continuously low, and when you momentarily press the button/tact switch, the output changes to continuously high, and then when you press it again, it switches back to low. install other packages when creating new package in R