I'm looking for a proof verification, precisely at the end of the proof. We proceed by induction on $n$.  Pf. Base case $n = 1$,$$\sum_{j=0}^1 {1 \choose j} = {1 \choose 0} + {1 \choose 1} = 1 + 1 = 2^1$$ Notice also for $n=2$, $$\sum_{j=0}^2 {2 \choose j} = {2 \choose 0} + {2 \choose 1} + {2 \choose 2} = 1 + 2 + 1 = 2^2$$ Now, suppose that for $k\in\mathbb{N}$ it is true that $$\sum^k_{j=0} {k \choose j} = 2^k$$  Then, by Pascal's identity,  $$\sum^k_{j=0} {k \choose j} + \sum^k_{j=0} {k \choose j-1} = \sum^{k+1}_{j=0} {k+1 \choose j} = 2^{k+1}$$ Since $k$ is any natural number, it should then also work for $k+1$.  My reservations are that I'm using the assumption incorrectly or misusing the assumption by not starting with the assumption. Any clarity is appreciated.  Edit: It seems that changing the bound on the second sum in this equation     Then, by Pascal's identity,    $$\sum^k_{j=0} {k \choose j} + \sum^k_{j=0} {k \choose j-1} = \sum^{k+1}_{j=0} {k+1 \choose j} = 2^{k+1}$$   to $k+1$ yielded the quickest fix to the original proof, with the convention that, as Brian M. Scott pointed out: ${n \choose k}$ is 0 if $k$ is a negative integer.
Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction.
I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk? I am trying to animate in 2d. I do it with planes that have textures on them. I recently came up with the idea of having a plane with an image sequence of mouths because animating actual lip syncing by hand takes too long and isn't really necessary for what I'm trying to acomplish. However, when playing back the animation in textured viewport shading the animated texture stays at the last frame until the animation stops. I can use the arrow keys and it works fine as well as scrubbing through the timeline, just not when playing back the animation with alt+a. Can anyone help me with this problem? How do I resolve unmet dependencies after adding a PPA? Occasionally, when I'm installing stuff, I get an error like the following:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  package1 : Depends: package2 (&gt;= 1.8) but 1.7.5-1ubuntu1 is to be installed E: Unable to correct problems, you have held broken packages.   How can I resolve this? I'm working on exercises from Kunen and I'm stuck. I must proof that the following are equivalent:   There exists a sequence $\langle A_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha)$ and for all $A \subset \omega_1$ the set $\{\alpha \in \omega_1:A\cap\alpha=A_\alpha\}$ is stationary. ($\diamondsuit$-principle) There exists a sequence $\langle A_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha\times\alpha)$ and for all $A \subset \omega_1\times\omega_1$ the set $\{\alpha \in \omega_1:A\cap\alpha\times\alpha=A_\alpha\}$ is stationary. There exists a sequence $\langle f_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(f_\alpha:\alpha\rightarrow\alpha)$ and for all $f : \omega_1\rightarrow\omega_1$ $\exists\alpha(f|\alpha=f_\alpha\wedge\alpha&gt;0)$ There exists a sequence $\langle f_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(f_\alpha:\alpha\rightarrow\alpha)$ and for all $f : \omega_1\rightarrow\omega_1$ the set $\{\alpha \in \omega_1:f|\alpha=f_\alpha\}$ is stationary.   I have proved that $1\implies 2\implies 4\implies 3$ and that $4\implies 1,$ but I'm having problems on proving that $3\implies$ "something", because it only talks about one $\alpha$. Can someone help me? How to add repository from shell in Debian? How can I formalize that the function $$f:\mathcal{P}(A)\to(A\to \{0,1\})$$ is bijective? It seems clear to me but I don't know how to write a proof. Thanks in advance. QM without complex numbers I created a simple service layer polygon in network analyst (ArcGIS Desktop 10.2) with no hierarchies set in the street network. Now I don't understand why is this polygon created along some particular streets?   Why isn't is properly spread out? For example, why doesn't the polygon cover the areas/streets I have marked in red? Obviously those marked areas are also within 1 km.  The impedance is 1 km and 2 km respectively.   The streets are also well connected individual line elements as you can see the junctions. How do we control web page caching, across all browsers? Our investigations have shown us that not all browsers respect the HTTP cache directives in a uniform manner.  For security reasons we do not want certain pages in our application to be cached, ever, by the web browser. This must work for at least the following browsers:   Internet Explorer 6+ Firefox 1.5+ Safari 3+ Opera 9+ Chrome   Our requirement came from a security test. After logging out from our website you could press the back button and view cached pages. How can I change (or remove / reset ) my name on a site in Stack Exchange network? Can I use my laptop as a second monitor? Is it possible to make my laptop's screen act as a second monitor for my desktop computer? Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? Are fractions with prime numerator and denominator dense? So the question is in the title. And I mean dense in positive real numbers ofcourse. Somehow I cannot grasp if this is very trivial or not. The prime numbers aren't that dense, but are there enough of primes to make the fractions constructed from them dense? Let  $$0 \rightarrow M \rightarrow N \rightarrow P \rightarrow 0$$  be an exact sequence of $A$-modules, where $A$ is a commutative ring with unity.  Then for each natural number $i$ one can look at the induced maps  $$\mathrm{Sym}^i(N) \rightarrow \mathrm{Sym}^i(P)$$  and   $$\mathrm{Sym}^i(M) \rightarrow \mathrm{Sym}^i(N).$$  The first question is, if they are still surjective, resp. injective, i.e. is $\mathrm{Sym}^i$ an exact functor?  The second question is: what are the kernel, resp. cokernel of these maps? How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? What are the most accepted ways to visualize the results of an independent two sample t-test? Is a numeric table more often used or some sort of plot? The goal is for a casual observer to look at the figure and immediately see that they are probably from two different populations. How can I launch screensaver with a keyboard shortcut? I want to shade the area of overlap for adjacent normal curves, shown in the lower graph. I can get shading between the first two curves but not for subsequent curves. I need shading at overlapping areas indicated by the arrows. I applied the "logic" of the first fill for the subsequent curves but that does not work.   The arrows and letters in the example image are not part of the MWE.   \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{ticks=none} \pgfplotsset{compat=1.7}  \usepgfplotslibrary{fillbetween}  \pgfmathdeclarefunction{gauss}{2}{%normal distribution where #1 = mean and #2 = sd}     \pgfmathparse{exp(-((x-#1)^2)/(2*#2^2))}% }  \pgfplotsset{baseplot/.style={%     no markers,      domain=1:4.5,      samples=100,     smooth,     axis lines*=left,     height=5cm, width=12cm,     enlargelimits=upper, clip=false, axis on top,     xlabel = near ticks,     xlabel={Resource} }}  \begin{document}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(2.7,0.4)};   \addplot+[name path=B]{gauss(3.2,0.4)};   \fill[gray!20, intersection segments ={of= B and A}]; \end{axis} \end{tikzpicture}  \vspace*{3\baselineskip}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(1.7,0.15)};   \addplot+[name path=B]{gauss(2.2, 0.15)};   \addplot+[name path=C]{gauss(2.7, 0.15)};   \addplot+[name path=D]{gauss(3.2, 0.15)};   \addplot+[name path=E]{gauss(3.7, 0.15)};   \fill[gray!20, intersection segments ={of= B and A}];   \fill[gray!20, intersection segments ={of= C and B}];   \fill[gray!20, intersection segments ={of= D and C}];   \fill[gray!20, intersection segments ={of= E and D}]; \end{axis} \end{tikzpicture}  \end{document} What does eno mean in network interface name 'eno16777736' for CentOS 7 or RHEL 7? Under consistent network device naming scheme, what does 'eno' stand for in network interface name eno16777736 for CentOS 7 or RHEL 7? How do I combine standard deviations of two groups? I have 2 groups of people. I'm working with the data about their age. I know the means, the standard deviations and the number of people. I don't know the data of each person in the groups.  Group 1 :  Mean = 35 years old; SD = 14; n = 137 people  Group 2 :  Mean = 31 years old; SD = 11; n = 112 people  I want to combine those 2 groups to obtain a new mean and SD. It's easy for the mean, but is it possible for the SD? I do not know the distribution of those samples, and I can't assume those are normal distributions. Is there a formula for distributions that aren't necessarily normal?