Problem  I am a software engineer and my friend and I are trying to pick up combinatorics. My friend asked me the following problem. How many ways to form 4 letter word from AAAADDDEEFGG  Attempt  My friend proposed $ 6 \choose 4 $. However, I felt this excluded the following possibilities:    Exclude A: $\frac{4!}{3!2!2!}$ Exclude D: $\frac{4!}{4!2!2!}$ Exclude E or G: $\frac{4!}{4!3!2!}$ Exclude F: $\frac{4!}{4!3!2!2!}$   However, when I sketched out the above I felt I was being rather silly since the answer was nonsensical. Please note that I based my approach off . I clearly think my answer is wrong.   Instead, should I exclude A and realize I have  $ 8 \choose 4$ ways to select from the remaining letters? If so, then I believe it be something like $\frac{8 \choose 4}{3!2!2!}$ when excluding A? The cases work similar for D, E, G, and F and I simply sum the results.   If not, can someone please give me a hint so I don't feel embarrassed with myself in the future :) Thanks.
How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm wondering if there's a simpler way to computing all the possible permutations, as an alternative to manually calculating all the possible 6-letter combinations and summing the corresponding permutations of each. In addition, is there a method to generalize the result based on any P number of letters in a set with repeating letters, to calculate the number of n-letter permutations?  Sorry if this question (or similar) has been asked before, but if it has, please link me to the relevant page. I also apologize if the explanation is unclear, or if I've incorrectly used any terminology (I'm only a high-school student.) If so, please comment. :-)
Is a function defined at a single point continuous? Is a function defined at a single point continuous?  For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for $f$ to be continuous, it has to have the limit $\lim\limits_{x_0\to x}f(x_0)=f(x)$ but it dosen't seem to have that limit. How to Flatten a Multidimensional Array? I have keypair (u,r), and two pieces of data, X and Y.  I reveal X publicly but keep Y secret. Y has high entropy.  I then concatenate X and Y, into Z.  I encrypt Z, into "encrypt(Z,u) = Q". r is needed to decrypt Q back into Z.  Later, (when it doesn't matter), r will be revealed and everyone will decrypt Q into Z. At that point, everyone will know that, all along, Q "contained" X. Can I prove that Q contains X before r is revealed (including interactive zero-knowledge proofs)? Is there an easy way to get to www.stackexchange.com from any of the SE sites? One thing I've noticed about the relatively new sitewide layout is that I can't find a quick way to get back to www.stackexchange.com without finding StackExchange in the sites list. As a result, I am typing stackexchange in my toolbar a whole lot more often now. In particular, I tend to enjoy browsing the hot questions list and my own personal feed quite often so visit these pages on a daily basis. Is there an easy link to them that I'm missing?  (On the android app, there is a feed in the menu that I really appreciate, something like that on the desktop would be ideal!) I managed to successfully publish my old bachelor thesis in a good quality national journal in my country. However, I am thinking of retracting the article due to certain circumstances. First, I felt really embarrassed by the low quality research methodology used in my bachelor thesis. I only used simple correlation in my research. After it was published, I realized that the journal editor team forgot to include a reference in that article because they were rushing me throughout the editing process. I informed them about their mistake and it seems that they have done nothing to fix the problem. I became a bit annoyed because the editor team has never made this problem in the past.  I felt that it would be better for me to simply ask for retraction to solve this issue, but I am a bit afraid of negative consequences in the future. Is it okay to do this? I am really interested of entering the academic world in the future. Are there any areas that should be avoided and are there any steps I should take to protect myself from radiation? Split a commit into smaller commits with Git? How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$? Define $f : [0,1] \to \mathbb{R} $ differentiable satisfying $f(0)=0$ and $0&lt;f'(x)&lt;1$. Prove:  $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$ I found the following answer,  Define $F(x)=\displaystyle\int_{0}^{x} f^3(t) \, dt - \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)^2$ We have $$F'(x)=f^3(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)f(x)=f(x)\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]$$ and $$\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]'=2f(x)\left[f'(x)-1\right]$$ from which we can arrive at the conclusion.  But I felt a little confused. In the past I usually proved the inequality by classical inequalities and something like Taylor expansion to use the condition about derivative. But this answer is little out of my expectation. Do we have other proofs or ideas to arrive at $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$? And other thoughts about solving these types of issues? I think when we talk about $\left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2 &gt;\ldots \,$ it's a little difficult to deal with. I don't quite know what tools to use. Why is the product of all units of a finite field equal to $-1$? Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by $a$. Then the product in question can be written as  $$ a_1\cdots a_{q-1}=1\cdot a\cdot a^2\cdots a^{q-2}=a^{(q-2)(q-1)/2}. $$ However, I'm having trouble jumping from that product to $-1$. What is the key observation to make here? Thanks! Find probability using distribution function method Prove 24 divides $u^3-u$ for all odd natural numbers $u$ At our college, a professor told us to prove by a semi-formal demonstration (without complete induction):   For every odd natural: $24\mid(u^3-u)$   He said that that example was taken from a high school book - maybe I din't get something, but I really have no idea to prove that without using complete induction.  Any idea of smart demonstration?  Thanks for your help. (Excuse my bad English.) Are there popular but poorly paved roads on Big Island which require an SUV? My grandparents will drive only on roads recommended in printed travel guides; they'll ask the rental car company if Waipio Valley Road and the road to the top of Mauna Kea are sanctioned.       A rented 4WD vehicle is allowed by law to travel ANY public roadway in Hawaii. AND there are some places that require a 4WD vehicle.      [...]      If you travel down into  or to the top of , you are legally allowed to do so, but your rental company will expect you to pay for any damage incurred whether you selected the insurance or not, as the insurance is null and void when you violate the contract.        I do not believe a 4x4 rental is necessary for 99% of visitors. For the other 1%, "necessary" is too strong a word!      But, being in that "1%" category, there ARE worthy places to go on the BI that require 4x4 (the kind you can rent from a normal rental outlet, not the kind locals may drive). I won't publicize those places because I don't want such places to become overused. [I bolded.] Just don't believe it when you read here that there is absolutely no place at all worth visiting that would require 4x4.    I doubt the Tripadvisor comments overhead, as none of them discussed potholes:          Please see the bolded phrase. Where on the BI requires an SUV? Old movie about a group of friends with magical weapons fighting monsters I need to find out the modular inverse of 5(mod 11), I know the answer is 9 and got the following so far and don't understand how to than get the answer. I know how to get the answer for a larger one such as 27(mod 392) but am stuck because they are both low numbers.  11=5 (2)+1  5=1 (5) I am using ArcGIS 10.1 and I have a non-spatial table and some layers in a Data Frame.   When I switch to Layout View I want to be able to show the map in a data frame element and the table in another element (something that one could do with "Add Table To Layout" from Table Options, but I want to do this in one step without having to go to Table Options).   Is there a way to do this using ArcPy?   I couldn't find any function in the ArcPy library that would create a new Layout Element ... and then I also don't know how to add the contents of the table to the Layout Element. Why XML-Serializable class need a parameterless constructor Define a sequence $(a_n)_{n = 1}^{\infty}$ of real numbers by $a_n = \sin(n)$. This sequence is bounded (by $\pm1$), and so by the , there exists a convergent subsequence.  My question is this: Is there any way to construct an explicit convergent subsequence?  Naïvely, I tried considering the subsequence $$\sin(3), \,\sin(31), \,\sin(314), \,\sin(3141),\, \sin(31415),\, \dots$$ hoping it would converge to $0$, but I was unable to prove this (and I think it's probably not even true). How do I find out which paths my (La)TeX installation checks looking for packages (on a Linux system)? I know that I can check from where one particular package would be called, using  kpsewhich &lt;packagename&gt;.sty   and that if I want a custom package, I usually have to put it in the same folder as my document.   I assume that similar to a $PATH, LaTeX checks a list of paths in an attempt to find my package.   Which paths are those? In what order are they checked? If this is system-specific, how can I find out the configuration for my system?   Related questions:    , the answers to which contain some paths that  LaTeX apparently checks.  mentions    &gt;    /usr/local/texlive/2009/texmf &gt;    /usr/local/texlive/2009/texmf-dist &gt;    /usr/local/texlive/texmf-local &gt; &gt; as well as other directories.   (Neither of those lists is complete). fetching list of friends using facebook 2.0 Tracking down an old sci-fi book I'm trying to track down an old sci-fi book I read when I was a teenager.  The book was at the Cottage Grove, OR Public Library around 1982-1985.  The plot followed a brother and sister who were scientists or spies, or both.  The brother once used a series of pills to change his weight, height and appearance.  Towards the end of the book, the brother takes a space ship that his sister invented, removes a governor so that he can travel faster than the speed of light and goes back in time several hours.  This is really vague, but thanks for any leads you can share.