It is proposed to colour the squares of a $4 \times 4$ board black and white, so that there are exactly two black squares and two white squares in each row or column.   In how many ways can this be done?  I've tried this finding out the number of ways column 1 and 2 can be done which is $36 = 6 \cdot 6$ as $\binom{4}{2}$ since the second column doesn't get in the way of the first.  Then I hit a road block. Can anyone provide a hint?
I don't know if this is a famous problem but my professor assigned this for a research project and I am hitting a dead end( I don't know what to do next).   Note: In $\phi(n)$ ; $n=rows=columns$ and this function throws out total combination of matrices that satisfies the following rule:  Rule for this square binary matrix:  Every Row and Column has two 1's.  So,    $$\phi(2)=1$$ \begin{bmatrix}    1 1 \\    1 1 \\ \end{bmatrix} $$\phi(3)=6$$  I can't write out all the possible matrices for $n=3$ but this is one of them (btw i did this by hand): \begin{bmatrix}    0 1 1 \\    1 0 1 \\ 1 1 0 \\ \end{bmatrix}   According to a program made by my friend, he told me:   $$\phi(4)=90$$ $$\phi(5)=2040$$
Website home page design when there are multiple audiences I am in the process of creating a website which has two distinct audiences - in my case consumers and retailers.  When I first started the process just a few weeks ago I had followed the then model used by PayPal who too have to deal with distinct audiences.    However, having Googled around a bit for what the accepted wisdom is when dealing with website landing pages for segmented markets it appears to me that the current belief is that one just directs the visitor away from the landing page to other, segment focused, pages ASAP.  I have to admit that I am having trouble buying that philosophy.  To my mind it makes sense to provide at least some basic information - key benefits - for each segment on the landing page and then let viewers who linger long enough the opportunity to easily drill down to their own segment-focused page.  My own reasoning is that most first time visitors are liable to want to bounce away quickly and making them find and click on the segment page link is more likely to make them bounce than if they find enough to hold them on the landing page for a bit longer.  I'd like to have other views on this.  What are the pros and cons of the    Drive traffic to segment-focused page ASAP vs Provide some basic information - at least key benefits - for each segment on the landing page itself Are Jedi bound by a code of honor or morals? In The Phantom Menace alone, Qui-Gon (and in part Obi-Wan) are quick to use Jedi mind tricks and not only gamble but also cheat at it. They seem happy with this fact.  Do Jedi have morals and ethics, or a code of honor? Disable the activation of the menu bar when Alt is pressed in Windows 7 In Windows 7 how can I disable the function that activates the menu bar when Alt is pressed?  Are there some registry values to modify this behaviour? In Windows 8, when Alt+Tab, the desktop would appear as an option to switch to at the end of the list of open programs (thus Alt-Shift-Tab helpfully took you right to the desktop).  In Windows 10 that desktop option is gone. Is there any way to get it back?  I know about Windows+D but Alt+Tab is just more convenient in my opinion. And  also has no desktop and is uglier. How can I search tags on StackOverflow Careers? I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). So what I decided to do is to start a small case with only 3 people. There are three possible combinations that I could pair up the people, using the symbols A, B, and C to represent the persons involved: A&amp;B, A&amp;C, B&amp;C. The probability of each pair having the same birthday is $1\over365$ (the first guy gets any birthday, and the second guy only gets one birthday to choose from). What my intuition directed me to do next is to find the probability that any of the three events are true, or the union of the three events, which involved adding up the probabilities.   To put this generally, the equation for the probability SHOULD be:  ${n \choose 2} \div 365$  However, clearly this is not the case since $n \choose 2$ gets over 365 when $n = 28$. And with 28 people, there is obviously still a chance that they all have different birthdays (Person 1 has Jan. 1, Person 2 has Jan. 2. ... Person 28 has Jan. 28). Could anyone tell me what's wrong with my intuition? I don't want to know what's the right solution, I just want to know what's wrong with my solution .. it makes sense to my intuition, even though it's incorrect. JavaFX accordion with multiple open panes How does Edo Tensei Madara get back his Rinnegan? Madara gave his Rinnegan to Nagato. But after Kabuto brings him back using Edo Tensei, he still has the Rinnegan. How is this possible? Control Cycles/Eevee material nodes and material properties using python? Get all non-unique values (i.e.: duplicate/more than one occurrence) in an array Prevent pdflatex from writing a bunch of files If I create just a basic hello.tex file an compile it using   pdflatex hello.tex   The pdflatex program is creating a plethora of file which is a little annoying. Currently it writes all those files about which I really don't care:  hello.aux, hello.log, hello.nav, hello.out, hello.snm, hello.toc   Is there a way to tell pdflatex to only write hello.pdf if the build is successful ? If not, can I specify the output directory of those files ? Maybe I could just write them in /tmp. I have a long line of code that I want to break up among multiple lines. What do I use and what is the syntax?  For example, adding a bunch of strings,  e = 'a' + 'b' + 'c' + 'd'   and have it in two lines like this:  e = 'a' + 'b' +     'c' + 'd' Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$  It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation? java swing background image Choosing whether to include PDF or PNG in PDFLaTeX Can I tell graphicx somehow which graphic format to choose when I omit the file extension in includegraphics and both a PDF and a PNG version of the same image exist? The reason is that I have some very large PDF vector graphics (in terms of file size/drawing primitives) that take a pretty long time to load each time I view the document. I would like to convert them to PNGs while working on the document and only use the PDF versions in the final document. Therefore, a global option that I can switch in the end would be great.  An additional plus would be to do the conversion to PNG on the fly during the LaTeX run if there is a PDF that has not been converted to PNG yet, if a tool for that exists. (I know how to use convert, but not how to include it automatically in LaTeX.) Mark my own question as favorite? Is there an idiom that means "revealing a secret unintentionally"? I am thinking there could be an idiom that states that you or someone revealed a secret unintentionally? Can you think of an idiom like that? I am looking for an idiom I can use to write a blog post about the recent incident where Secretary Pompeo unintentionally revealed the dirty secrets behind the CIA. What can I do to give the players the same feel their characters would have about wilderness travel? When I cut a layer using marquee tool, I get blur edges to cut side. How do I get sharp edges after cut? I am using rectangular marquee tool to cut a portion of the layer in photoshop cs5, but after cutting, the edges are not sharp, I get blur edges.      how can I get sharp edges? I am still learning settings API so please forgive my noob question. I have three fields: a checkbox, colorpicker and text field (they are working), but I would like to do this: when the theme is installed add "checked" state to my checkbox but allow user to "unchecked" it ,and if empty add default color to my colorpicker... I have no idea how to do this so please help me!  My code looks like this now (thanks to @Chip Bennett! ) :  &lt;?php add_action('admin_menu','create_theme_options_page'); function create_theme_options_page(){ add_options_page('Theme Options','Theme Options','administrator',__FILE__,'build_options_page'); }    function build_options_page(){    ?&gt;    &lt;div id="theme-options-wrap"&gt;    &lt;div style="float:left;margin-top:19px;position:absolute;" class="icon32" id="icon-tools"&gt;&lt;br/&gt;&lt;/div&gt;    &lt;h2&gt;My options page&lt;/h2&gt;    &lt;form method="post" action="options.php"&gt;    &lt;?php settings_fields('plugin_options');?&gt;    &lt;?php do_settings_sections(__FILE__); ?&gt;    &lt;p class="submit"&gt;    &lt;input name="Submit" type="submit" class="button-primary" value="&lt;?php esc_attr_e('Save changes');?&gt;"/&gt;    &lt;/p&gt;    &lt;/form&gt;    &lt;/div&gt;    &lt;?php    }       add_action('admin_init','register_and_build_fields');      function register_and_build_fields(){      register_setting('plugin_options','plugin_options','validate_setting');      add_settings_section('main_section','&lt;br/&gt;&lt;br/&gt;Main Settings','section_cb',__FILE__);      add_settings_field('banner_heading','Banner Heading:','banner_heading_setting',__FILE__,'main_section');      add_settings_field('display_sidebar','Display Sidebar:','display_sidebar_setting',__FILE__,'main_section');      add_settings_field('display_colorpicker','Display Colorpicker:','display_colorpicker_setting',__FILE__,'main_section');      }        //Install Iris colorpicker        add_action( 'admin_enqueue_scripts', 'mw_enqueue_color_picker' );        function mw_enqueue_color_picker( $hook_suffix ) {        // first check that $hook_suffix is appropriate for your admin page        wp_enqueue_style( 'wp-color-picker' );        wp_enqueue_script( 'my-script-handle', get_template_directory_uri() . '/js/my-script.js', array( 'wp-color-picker' ), false, true );        }           //Colorpicker          //Now how to add a default color on theme install?          function display_colorpicker_setting(){          $options = get_option('plugin_options');          echo"&lt;input type='text' id='color1' name='plugin_options[display_colorpicker]' value='{$options['display_colorpicker']}'/&gt;";          }            //Sidebar checkbox           //Now how to add a default state "checked" on theme install?           function display_sidebar_setting(){           $options = get_option('plugin_options');           echo "&lt;input type='checkbox' name='plugin_options[display_sidebar]' {$options['display_sidebar']} id='display_sidebar' /&gt;";           }              //Banner Heading             function banner_heading_setting(){             $options = get_option('plugin_options');             echo "&lt;input name='plugin_options[banner_heading]' type='text' value='{$options['banner_heading']}'/&gt;";             }               //Validate our settings here               function validate_setting($plugin_options){                  return $plugin_options;               }                 function section_cb(){                } ?&gt;