Permutations And Combinations

Question
CBSEENMA11014196

Determine n if:

straight C presuperscript 2 straight n end presuperscript subscript 3 space colon space straight C presuperscript straight n subscript 3 space equals space 11 colon 1

Solution

straight C presuperscript 2 straight n end presuperscript subscript 3 colon straight C presuperscript straight n subscript 3 space equals space 11 colon 1
rightwards double arrow      fraction numerator 2 straight n factorial over denominator left parenthesis 2 straight n minus 3 right parenthesis factorial end fraction colon space fraction numerator straight n factorial over denominator left parenthesis straight n minus 3 right parenthesis factorial space 3 factorial end fraction space equals space 11 colon 1


rightwards double arrow      fraction numerator left parenthesis 2 straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis space left parenthesis 2 straight n minus 2 right parenthesis thin space left parenthesis 2 straight n minus 3 right parenthesis factorial over denominator left parenthesis 2 straight n minus 3 right parenthesis factorial space 3 factorial end fraction cross times fraction numerator 3 factorial space left parenthesis straight n minus 3 right parenthesis factorial over denominator straight n left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 2 right parenthesis thin space left parenthesis straight n minus 3 right parenthesis factorial end fraction equals 11 over 1
rightwards double arrow            fraction numerator left parenthesis 2 straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis thin space left parenthesis 2 straight n minus 2 right parenthesis over denominator straight n space left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 2 right parenthesis end fraction equals 11 over 1
 
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#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>
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#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                                3n = 18
rightwards double arrow                                 n = 6
Hence,    n = 6.

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