Can someone explain how to prove the following identity involving Fibonacci sequence  $F_n$  $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ?
Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$;  I was stuck with this question for a while... Help me please!!! Thanks!!!
How do I get an up/parent button in Nautilus, without using breadcrumbs? US tourism/business visa from non-resident country Redirect STDERR and STDOUT to different variables without temporary files Recommended books for advanced undergraduate electrodynamics Remove empty elements from an array in Javascript How do I remove empty elements from an array in JavaScript?   Is there a straightforward way, or do I need to loop through it and remove them manually? A different approach in distributing $8$ distinct balls into $6$ distinct boxes Find the number of ways in distributing $8$ distinct balls into $6$ distinct boxes such that there is at least $1$ ball in each box.  We are well acquainted with the traditional Inclusion-Exclusion principle in solving this and it will come up as $191520$ ways.  I tried in this way:  When we distribute $8$ balls into $6$ boxes we have two possible cases  Case $1.$ There are exactly $2$ boxes with $2$ balls each  Case $2.$ There is exactly $1$ Box with $3$ balls in it.  For Case $1$, the distribution is done by choosing $2$ boxes from $6$ boxes and $2$ balls from $8$ balls and those two balls can be arranged in $2!$ ways. But I require another two balls from remaining $6$ balls to be placed in each of these two boxes which can be done in $\binom{6}{2} \times 2!$ ways. Finally remaining $4$ balls can be distributed in remaining $4$ boxes in $4!$ ways. So total number of ways for Case $1.$ is  $$\binom{6}{2} \times \binom{8}{2} \times 2! \times \binom{6}{2} \times 2! \times 4!=604800$$  For Case $2$, the number of ways is  $$\binom{6}{1} \times \binom{8}{3} \times 5!=40320$$  I am pretty sure the value obtained for Case $2.$ is correct. but I could not find where I went wrong in Case $1.$ Move item to folder when added/edited (SP2013 or O365) How variance is defined? The variance of a random variable $X$ is defined as $E[(x-\mu )^2]$. Why can't it be defined as $E[|x-\mu |]$. i.e., What is the basic idea behind this definition. Thank you. I have installed Apache(LAMP) web server on my ubuntu machine and I want to do the above mentioned thing.I have tried creating symbolic links in the /var/www directory but it only shows files but not directories inside the directory linked.I want to redirect /var/www to my home folder so that I can access all the files and directories contained therein.Please help me. What are some alternatives to "hover" on touch-based devices? Came up recently in a 37signals podcast:    The straightforward, "brute-force" solution is to just use the first touch on a UI object as the hover.    It works. But are there other, more elegant solutions? For instance, could the hardware ever be made to support actual finger hovering? Surjectivity implies injectivity and conversely Let S be a finite set.Let F be a surjective function from S to S.  How do I prove that it is injective? what could cause a script to fail to find python when it has `#!/usr/bin/env python` in the first line? Trying to get casperjs running on Ubuntu 12.04. After installing it when I run I get:   09:20 $ ll /usr/local/bin/casperjs lrwxrwxrwx 1 root root 26 Nov  6 16:49 /usr/local/bin/casperjs -&gt; /opt/casperjs/bin/casperjs  09:20 $ /usr/bin/env python --version Python 2.7.3  09:20 $ cat /opt/casperjs/bin/casperjs | head -4  #!/usr/bin/env python  import os import sys  09:20 $ casperjs : No such file or directory  09: 22 $ python Python 2.7.3 (default, Sep 26 2013, 20:03:06)  [GCC 4.6.3] on linux2   So Python is present and runnable, casperjs is pointing to the right place and it is a python script. But when I run it I get "No such file".   I can fix it by changing the first line of the casperjs python file from:    #!/usr/bin/env python   to:   #!/usr/bin/python   Result:   $ casperjs --version 1.1.0-DEV   I managed to fix it, but I'm wondering why it didn't work with #!/usr/bin/env python, since that seems to be a normal interpreter line. Do I have something configured wrong?   Here are the steps to get casperjs:   $ git clone git://github.com/n1k0/casperjs.git $ cd casperjs $ ln -sf `pwd`/bin/casperjs /usr/local/bin/casperjs $ casperjs : No such file or directory Ideal column width for paragraphs online There is nothing worse than coming across a website with paragraphs that span the width of your screen resolution. They are usually so hard to read, I don't even bother.  A lot of web designers use generic grid and column systems, but I don't feel these have been optimised for readability, they are more of design guide on general aesthetics.  I guess it comes down to words per line. Are the rules that apply to print media the same for screen media?  Are there any type of guidelines to help me size the width of paragraphs on the screen?  I want to take into consideration:   Major web font faces (i.e. Arial, Verdana, Georgia, Times) A range font sizes (i.e. 8px-24px) Space between columns Explanation behind Second Derivative of a Parametric Equation Formula I am looking to find out how the second derivative formula works.  I can blindly apply it but I don't have a grasp of what is going on or why. The first derivitive has the formula $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. I told this was because if you divide them you cancel out the dts and get dy/dx. The second derivative has the formula $\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{ \frac{d}{dt}\left(\frac{dy}{dx}\right) }{\frac{dx}{dt}}$. Is this something that can be explained with only a knowledge of calculus or is this something I would have to take on faith until I learn more. Any help would be greatly appreciated. What influences findability on a post?   Upvotes Views Asked date Last active time   How is this done in the Stack Exchange internal search engine? And how does Google do it?  Why is it that when searching on Google for 'iPhone development stackoverflow' the fourth result is ? I'm arriving to the US by air, but departing via land. How do I end form I-94 now that it is only electronic? The company I work for has asked me to attend a training class on the other side of the country.  However my situation is my drivers license expired a few days ago. I do however have a separate printed email stating that the license has been renewed.  This drivers license is my only ID that shows both my name &amp; photo.  Is it possible to be able to board a flight with the expired drivers license and the printed email showing that it was renewed? How can one predict whether an individual infected with the SARS-CoV-2 virus will be asymptomatic? According to  , half of the   virus carriers show no symptoms.   What are the biological factors that determine whether an infected individual will be asymptomatic or not? How to show $\omega^n$ is a volume form in a symplectic manifold $(M, \omega)$? Is history expansion disabled in scripts? I understand the following error is due to ! used for history expansion:  $ echo "Hello!Tim" bash: !Tim: event not found   However if I put the command into a script and run the script, there is no problem:  $ cat myscript echo "Hello!Tim" $ bash myscript  Hello!Tim   Why is that? Does the bash manual mention the reason?