Permutations and Combinations

Question

Find r, if

$space straight P presuperscript 5 subscript straight r space equals space 2. straight P presuperscript 6 subscript straight r minus 1 end subscript$

Solution

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $fraction numerator 5 factorial over denominator left parenthesis 5 minus straight r right parenthesis factorial end fraction space equals space 2. space fraction numerator 6 factorial over denominator left parenthesis 6 minus straight r plus 1 right parenthesis factorial end fraction$
$rightwards double arrow$ $fraction numerator 5 factorial over denominator left parenthesis 5 minus straight r right parenthesis factorial end fraction space equals space 2. fraction numerator 6 factorial over denominator left parenthesis 7 minus straight r right parenthesis factorial end fraction$
$rightwards double arrow$ $fraction numerator 5 factorial over denominator left parenthesis 5 minus straight r right parenthesis factorial end fraction space equals space 2. fraction numerator 6 cross times 5 factorial over denominator left parenthesis 7 minus straight r right parenthesis space left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis factorial end fraction$
$rightwards double arrow$ $1 equals fraction numerator 2 cross times 6 over denominator left parenthesis 7 minus straight r right parenthesis space left parenthesis 6 minus straight r right parenthesis end fraction$
$rightwards double arrow$ $space space straight r squared minus 13 straight r plus 42 space equals space 12$
$rightwards double arrow$ $space space straight r squared minus 13 straight r plus 30 equals 0$
$rightwards double arrow$ (r - 10) (r - 3) = 0
$rightwards double arrow$ r = 10, r = 3
But, for $straight P presuperscript straight n subscript straight r comma space 0 space less or equal than space straight r space less or equal than space straight n$
But, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ [Not valid]
Hence, only value of r = 3

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Permutations And Combinations

Find r, if

$space straight P presuperscript 5 subscript straight r space equals space 2. straight P presuperscript 6 subscript straight r minus 1 end subscript$

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