Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362456 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0679 1211800 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0679 1211800 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1384 4039904 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1400 4067456 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1409 4085168 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1409 4102176 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1418 4109912 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1418 4109912 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1847 4339912 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1861 4357312 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1867 4378240 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1957 4444632 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1957 4444888 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Combinatorics and Factorials

If we have to choose a group of 6 people from a selection of 10, there are

possible ways of making a choice of six. The order of selection doesn't matter here. So that although each of the 10 are distinct, once they are picked they can line up in any order, and the order of lining up does not matter.

Working from first principles we might explain like this. We are going to pick 6 from ten. There are 10 possible choices for the first one, 9 possible choices for the second one, 8 for the third, 7 for the fourth, 6 for the fifth and 5 for the sixth, hence 10*9*8*7*6*5 possible ways for picking six altogether. We can write:

(1)

Once we have picked our selection the order in which they are lined up does not matter. For the six we have selected there aredifferent ways of lining them up. To take this into account we can divide (1) by an extra factor of 6! to obtain

possible ways of picking 6 from ten. This is of course the same as calculated above.

In general then there areways of choosingfromwithout regard to order. It should also be noted that