I'm struggling to prove $$k\binom{n}{k}=n\binom{n-1}{k-1}$$   Here's what I'm doing:  $$k\binom{n}{k}=\frac{k\cdot n!}{k!(n-k)!}=\frac{k\cdot n\cdot(n-1)!}{k(k-1)!(n-k)!}=n\frac{(n-1)!}{(k-1)!(n-k)!}$$  I see the problem that in the denominator we have $(n-k)!$ I guess it should be $(n-(k-1))!=(n-k+1)!$ to make the whole fraction equal $n\binom{n-1}{k-1}$  What's the problem?
I've seen many posts regarding a combinatorial proof of the following question. But for a non-combinatorial proof would the following method work? Also... is this the easiest way to arrive at a proof? It seems to be rather verbose.  Show the formula $k {n\choose k} = n {n-1 \choose k-1}$ is true for all integers $n,k$ with $0\le k \le n$.  My answer:  Observe that  $$\begin{align*} {n\choose k}&amp;= \frac {n}{k} {n-1 \choose k-1}\\ &amp;= \frac {n}{k}\left[ {n-2 \choose k-2} + {n-2 \choose k-1}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-2-(k-2))!} + \frac {(n-2)!}{(k-1)!(n-2-(k-1))!}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-k)!} + \frac {(n-2)!}{(k-1)!(n-k-1)!}\right]\\ &amp;= \frac{(n-1)!}{(k-1)!(n-k)!} + \frac {(n-1)!}{(k)!(n-k-1)!}\\ &amp;= \frac{(n-1)!}{(k-1)!(n-1-(k-1))!} + \frac {(n-1)!}{(k)!(n-1-k)!}\\ &amp;={n-1 \choose k-1} + {n-1 \choose k}={n \choose k}. \end{align*}$$
Fourier transform of the indicator of the unit ball Instead of relying on my host to send an email, I was thinking of sending the email messages using my Gmail account. The emails are personalized emails to the bands I play on my show.   Is it possible to do it? I have some files with different extensions such as *.pdf, *.mp3, *.jpg and a few others. All of them are stored in a parent directory.  How can I get a list of all extensions, create some folders based on these  extensions and then move all files into their relevant folders? Imagine that you have the following list. keys = ['name', 'age', 'food'] values = ['Monty', 42, 'spam']  What is the simplest way to produce the following dictionary? a_dict = {'name': 'Monty', 'age': 42, 'food': 'spam'} forecast using arima models I am trying to predict values using arima(0,1,1). After doing predict(mod,n.ahead=5) (in R) am getting the same value for all the predictions:   5947.681 5947.681 5947.681 5947.681 5947.681    Is it correct? getting exception while putting list data into the map Use biblatex and utf8 For my thesis, I use biblatex for the bibliography.  As seen , biblatex is not compatible, with \usepackage[utf8x]{inputenc}. It was reported as close but I still have the same problem with my version of bibtex (last one on archlinux ). I want to use the € symbol inside my report but it fails without utf8.  ! Package biblatex Error: Incompatible package 'ucs'.  See the biblatex package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.7 \begin{document}   Any idea idea how to fix ?  \documentclass{report}  \usepackage[utf8x]{inputenc} \usepackage{biblatex} \bibliography{includes/mybib}{}  \begin{document}  X said Y \cite{bib1}  \printbibliography  \end{document}   and the file mybib  @online { bib1,        title = {Refrence title},        author = {X},        url = {http://site.com/y}} Let $R$ be a commutative ring.  The nilradical $\text{nil}(R)$ is the set of all nilpotent elements, and it is the intersection of all the prime ideals of $R$.  Is the following true in the polynomial ring? $$\text{nil}(R[X])=(\text{nil}(R))[X]$$  Any polynomial in the right hand side has all its coefficients nilpotent in $R$, hence also in $R[X]$, and therefore it is nilpotent in $R[X]$ as the sum of nilpotent elements.  The other inclusion is not clear to me.  Is there a counterexample? Symmetric Inequality in $\mathbb{R}$ Modify all bash commands through a program before executing them I am trying to create a program which require such feature. The flow will be like:   User enter a bash command User hit the enter My script will get command, current directory,.. as variables. Program can optionally modify the command. Modified command will get executed normally.   Is there any way to do this?  Note: I need this for my personal use, I am not going to distribute this program. How do I counter-knife an enemy? When you press down the right stick to counter the attack, do you just press it, repeatedly press it, or press and hold? I've been knifed a few times from the front and each time I try to counter I fail. Also, if you believe you are about to be knifed and hold down the right stick until you are knifed, will that counter an attack, or do you have to press it down just at the right time? Filling in new field in attribute table based on values of another field in ArcGIS Desktop? I have added a new field (Category) in an attribute table of a shapefile and I would like to fill it with numbers, 0 or 1, based on the numbers of another field (Calorific value) of the same attribute table.   That means that if the Calorific value is higher than 15, then I would like to fill in the corresponding cells of the (new) Category field with 1. If the Calorific value is less than 15, then the rest cells of the (new) Category field to be filled in with the number 0.   Is there a way to do this automatically rather than manually? Automatically turn on VPN on computer unlock Ubuntu 18.04 When I wake the laptop from the sleep I need to get the VPN state as I left it (turned on). Can't find the setting of autoconnect in VPN settings... Need some easy way to ask the system to autoconnect the VPN when the internet is available.  Edit based on comments:  Ubuntu 18.04 has no option to Always connect to VPN when using using this connection. Windows 8 opens notepad on boot- no startup entry found I just upgraded from windows 7 to windows 8 and everything is working fine except notepad keeps opening on boot with a "access denied" window that comes up as well. I've ran MBAM and removed some junk, I've disabled every "file not found" item in autoruns as well as something that had a text/xml title in it and I've ran CCleaner a few times. Nothing is working, any help is much appreciated! How long is it safe to marinate meat? Proving $\arg(zw)=\arg(z)+\arg(w)$ This is my attempt I know this is incomplete or may even be wrong.   Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient for the proof or correct at all? Hope someone could help me out. Thanks.   EDIT: Just to avoid any confusions I will add how $\arg z$ is defined.   $\arg z=\{\text{Arg}\, z+2\pi k\mid k\text{ is an integer}\}$ How to compare (probability) predictive ability of models developed from logistic regression? I know some well-known measures are $c$ statistic, Kolmogorov-Smirnov $D$ statistic. However, as far as I know, those statistics take into account only of the rank order of the observations, and is invariant under changing the intercept of the logistic regression model (e.g. in oversampling-correction exercise).  In my current application, I need to depend on the accuracy of the logistic regression to predict probability of event. I know only of qualitative way of assessing models for probability prediction ability, namely by plotting "QQ-plot" of the actual vs predicted probability of event:   Score the validation dataset using the developed model. Rank the observations according to the predicted probability and group into $n$ buckets according to their rank of predicted probability. (First 1/n would go to the first bucket, next 1/n would go to the next ...) Calculate the average predicted and actual probability of Event for each bucket. Create a scatter plot of Predicted vs Actual - one point for each bucket.   I am wondering:   Is the "Q-Q plot" I mentioned above a legitimate way to assess predictive performance of models developed from logistic regression? If so, where may I find more reference for that? Is there any known quantitative way to assess the probability prediction ability of this kind of model? The entire answer is moving when hovering the bounty button It's just a minor CSS issue. See the following gif: Why absence of electron is called hole? Is "I'm being angry" ungrammatical?