Permutations and Combinations

Permutations and Combinations

Question

Find four numbers in A.P. whose sum is 32 and the product of extremes is to the product of means is 7 : 15.

Answer

Let the four number in A.P. be a - 3d, a - d, a + d, a + 3d.
Since the sum of four numbers  = 32

∴   a - 3d + a - d + a + d + a + 3d = 32
or                               4a = 32                         or                 a = 8
Since the ratio of the product of extremes to the product of means is 7:15.

∴      fraction numerator left parenthesis straight a minus 3 straight d right parenthesis left parenthesis straight a plus 3 straight d right parenthesis over denominator left parenthesis straight a minus straight d right parenthesis left parenthesis straight a plus straight d right parenthesis end fraction space equals space 7 over 15            or                 fraction numerator straight a squared minus 9 straight d squared over denominator straight a squared minus straight d squared end fraction space equals space 7 over 15
or         fraction numerator 8 squared minus 9 straight d squared over denominator 8 squared minus straight d squared end fraction equals 7 over 15                   or             space space 15 space left parenthesis 64 minus 9 straight d squared right parenthesis space equals space 7 space left parenthesis 64 minus straight d squared right parenthesis
or           space space space 960 minus 135 straight d squared space equals space 448 space minus space 7 straight d squared  or             <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>
or                <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>                       or                      straight d equals plus-or-minus 2
When d = 2 and a = 8 numbers are 8 - 3(2), 8 - 2, 8 + 2, 8 + 3(2)
                              2, 6, 10, 14
When d = -2 and a = 8 numbers are 8 - 3(-2), 8 - (-2), 8 + (-2), 8 + 3(-2)
                               14, 10, 6, 2

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