This is a question from Cambridge, Downing sample test     Using the definition $\left( \begin{array}{c}  \text{n}\\  \text{k}\\ \end{array} \right) =\frac{\text{n!}}{\text{k!}\left( \text{n}-\text{k} \right) !} $   , prove that $ \sum_{\text{j}=0}^{\text{n}}{\left( \begin{array}{c}  \text{n}\\  \text{j}\\ \end{array} \right) =2^{\text{n}}} $   Of course I know how to prove this by considering the subsets, or by binomial expansion.  But how to do it by definition? No idea about it.
Prove that$$\overbrace{\binom n 0 + \binom n 1 + \binom n 2 + \cdots + \binom{n}{\frac{n-1} 2}} + \overbrace{\binom n {\frac{n+1} 2} + \cdots + \binom n {n-1} + \binom n n} = 2^n. $$   Background:  This problem came from the following link:    and I didn't understand why that was true in which no one on the page has proved it to be so. and i was wondering what was the proof on that
Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$. $(function(){   $('.slider').mobilyslider({     content: '.sliderContent',     children: 'div',     transition: 'horizontal',     animationSpeed: 500,     autoplay: true,     autoplaySpeed: 9000,     pauseOnHover: true,     bullets: true,     arrows: true,     arrowsHide: true,     prev: 'prev',     next: 'next',     animationStart: function(){},     animationComplete: function(){}   });    });   I am integrating the above JavaScript file in a Drupal theme, but it is giving me the "$ is not function" error. I googled it, and changed the code bro the below, but it doesn't work either.  (function($) {   $('.slider').mobilyslider({     content: '.sliderContent',     children: 'div',     transition: 'horizontal',     animationSpeed: 500,     autoplay: true,     autoplaySpeed: 9000,     pauseOnHover: true,     bullets: true,     arrows: true,     arrowsHide: true,     prev: 'prev',     next: 'next',     animationStart: function(){},     animationComplete: function(){}   });    })(jQuery);   The above code gives me following error:     $('.slider').mobilyslider is not a function Question is about Ubuntu release process. Are migrations from n to n+1 releases expected?  Assuming we have server with installed Ubuntu 14.04 LTS. Will it be smoothly upgradable to Ubuntu 16.04 LTS via apt-get upgrade ?  Assumptions:   scope is software therefore we assume that given hardware is supported on both releases (upgrading from and to) that's why, always before upgrade you are asked to make backup to be able to revert in case of failure In most electrical circuits with a relay, a diode is connected in parallel to the coil of the relay. Why? Is it always a good practice? Relation between confidence interval and testing statistical hypothesis for t-test It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's assume that such hypothesis is tested using t-test. On the other side, one can compute confidence intervals for means of both groups. Is there any relation between overlapping of confidence intervals and the rejection of the null hypothesis that means are equal (in favor of the alternative that means differ - two-sided test)? For example, a test could reject the null hypothesis iff the confidence intervals do not overlap. Can a service written in .NET self-terminate? I have a service application written in C# and under certain circumstances, I would like it to terminate itself.  This would happen after the service has been running for a while, so this would not be happening in the OnStart() event.  Everything that I have read so far suggests that the only safe way to terminate a service is through the Service Control Manager.  My service runs as Local Service and does not have the rights to start or stop services, so I can't access the SCM from the service itself.  Is there another way to self-terminate while still playing by the rules of the SCM? How do moving charges produce magnetic fields? Is there a word for when a statement is technically true but misleading? End of File in C++ What invisible flicker do different types of light bulbs have? What waveform does the light coming off different types of artificial light sources have and (of course) what are the underlying physical reason for them?    I was inspired to ask this question because of several things I have been hearing lately.  Firstly, there was a recent TED talk about how .  Researchers have been working on the problem for a little while now and they've made a fairly impressive product, but it was only recently made possible by the LED technology, which has such a fast response that it can easily compare to the transfer rates of electric information condiuits that we normally use.  My understanding is that incandescent bulbs flicker because they have 60 Hz AC power being pulsed through them.  In fact, I think 60 Hz was chosen in the first place because it was a threshold that was unnoticeable to humans.  Looking at a  demonstrates that it has a very significant lag time responding to both the normal electrical phasor and being turned off.  In other words, it flickers but it doesn't go dark at any point in the AC waveform.  Looking for more information, however, I was a little disappointed, since nothing seemed to give a graph of lumens emitted over time for all different kinds of bulbs.  Of course, there isn't just one relevant time scale.  A light could flicker with 60 Hz and also flicker with 2000 Hz (I just made that up).  I'm interested in the complete picture of what lights exhibit what patterns. Not able to install Win Xp on Lenovo Ideapad z570 I want to install Win Xp on my Ideapad Z 570 laptop but i always get blue screen with message with some drive related problem which asks me to run CHKDSK command. I tried with that but no luck. Is there any way i can install Win Xp with SP 2 ?  Edit: i searched across the sites and came to know there could be some Compatibility problems, and i could fix it by setting HDD mode to IDE; will it solve the problem? I'm very familiar with how symbols display differently in inline mode (with $ ... $) vs. how display math shows them (with \[ ... \] or $$ ... $$).  Two examples would be with limits  $\lim_{n\rightarrow \infty}f(x)$   vs.  \[ \lim_{n\rightarrow \infty}f(x) \]   and with sums  $\sum_{n=1}^{x} n^2$   vs.  \[ \sum_{n=1}^{x} n^2 \]   My Question is:     How do you display these things in inline mode (with all the nice inline formatting) as if they were in display mode ?   The inverse -- how to display things in display mode as if they were in inline mode -- is also of interest. Number or regions formed when $n$ points on a circle are joined The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$.  This is a fact that I have read in several essays on the dangers of jumping to conclusions in mathematics.  In your opinion, what's the quickest (and/or nicest) way to prove this formula?  Professor  explains somewhere an especially convincing way to tackle this pearl following an idea which a certain high schooler in one of his talks (lectures?) came up with; yet, I don't remember  where it was that I actually read this...  Let me thank you in advance for your insightful replies. 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? Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$g(X) = 2$ and deg$h(X) = 3$) Recall that $\bar{a}$ is shorthand for the coset $a + 2\mathbb{Z} \space $in$ \space (\mathbb{Z}/2\mathbb{Z})[X]$ (for any $a \in \mathbb{Z}$)  This question is worth $7$ marks out of a possible $75$, this is my solution..  $f(X) = X^5 + X^3 + \bar{1}$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$ iff it has no roots in $\mathbb{Z}/2\mathbb{Z}$. Since $f(\bar{0}) = \bar{1}$ and $f(\bar{1}) = \bar{1}$, it follows that $f(x)$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$.  Would this be a sufficient answer? am I missing something? The exact phrase from the book of Swan (the context is choosing the words permit or allow) -      The passive structure with it is only possible with permit   Having said that...     It is not allowed to smoke in the kitchen - incorrect     It is not permitted to smoke in the kitchen - correct   I feel absolutely fine for the incorrect sentence.   Your inputs please. Make me understand this. Inverse of $y = x^3 + x $? Can you help me find the inverse function for $y = x^3 + x$?  This question was posed at the beginning of AP Calculus, so we can't use any math beyond precalc.  Thanks! How to get PHP 7.0 running with Apache2 I am totally new to Linux, servers and websites. I recently set up Apache2, Python, PHP, and MySQL for setting up a LAMP server. I am using Xubuntu 16.04 and PHP 7.0.11. I followed a guide and placed a PHP file named testing.php in it with:  &lt;?php phpinfo(); ?&gt;    It doesn't show anything on localhost/testing.php All it shows is the code itself. Terminal icon in window titlebar is stretched out over window control buttons A while ago I upgraded from Ubuntu 17.04 to Ubuntu 17.10 and everything is working fine except for the fact that the icon in the window title bar of my Terminal seems to be stretched out to some ridiculous size.    This only seems to happen in the Terminal window. The Nautilus icon is completely fine.    The problem seems to go away if I login into an Xorg session instead of a Wayland one. I've looked around and didn't find any immediate solution to this. Any help would be greatly appreciated. Man clones himself to have the clone do things he doesn't enjoy I read a short story about four years ago and can't remember the title or author. The only thing I know is that it is relatively old -- like Huxley or Asimov era. Can anyone help me?   The basic synopsis is this: a man clones himself and has his clone do the parts of his life that he doesn't enjoy. He deceives his wife about this; she doesn't know he has a clone. Whenever the man and the clone are both home, the clone is locked in the "toolbox" in the basement. The clone starts to treat the man's wife better than he had previously. His wife starts to fall in love the clone, and the man decides he needs to get rid of the clone. When he tries to do so, the clone overpowers the man and locks him in the toolbox in the basement!  I'm hoping to share this story with my English class this school year. Any help is appreciated!