Permutations and Combinations

Permutations and Combinations

Question

Find the sum to n terms of the following series.

space space space 1 cross times 2 cross times 3 plus 2 cross times 3 cross times 4 plus 3 cross times 4 cross times 5 plus........

Answer

Here, straight T subscript straight n equals end subscript [nth term of 1, 2, 3,......] cross times [nth term of 2, 3, 4,.....] cross times [nth term of 3, 4, 5......]
            = [1 + (n - 1). 1] [2 + (n - 1). 1] [3 + (n - 1) .1]
            = <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
Letspace space straight S subscript straight n denote the sum of n terms of series, then  
              straight S subscript straight n space equals space sum from straight k equals 1 to straight n of straight T subscript straight k space equals space sum from straight k equals 1 to straight n of left parenthesis straight k cubed plus 3 straight k squared plus 2 straight k right parenthesis space equals space sum from straight k equals 1 to straight n of straight k cubed plus 3 space sum from straight k equals 1 to straight n of straight k squared plus 2 sum from straight k equals 1 to straight n of space straight k
                     
                  = open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets squared plus 3. space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus 2. space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction plus fraction numerator 3. space left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 end fraction plus 2 close square brackets
equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis plus 2 left parenthesis 2 straight n plus 1 right parenthesis plus 2 cross times 2 over denominator 2 end fraction close square brackets
equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n squared plus straight n plus 4 straight n plus 2 plus 4 over denominator 2 end fraction close square brackets space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator straight n squared plus 5 straight n plus 6 over denominator 2 end fraction close square brackets
space space equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis straight n squared plus 3 straight n plus 2 straight n plus 6 right parenthesis over denominator 4 end fraction
<pre>uncaught exception: <b>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</b><br /><br />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<br />#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>
            

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