Permutations and Combinations

Permutations and Combinations

Question

Find two positive numbers whose difference is 12 and whose A.M. exceeds the GM. by 2.

Answer

Let a and b be two positive numbers such that
                              a - b = 12                                                                                                  ...(i)
Since  A.M. exceeds G.M. by 2

∴                fraction numerator straight a plus straight b over denominator 2 end fraction space equals space square root of ab plus 2            rightwards double arrow    a + b = 2 square root of ab plus 4
rightwards double arrow         straight a plus straight b minus 2 square root of ab space equals space 4                rightwards double arrow      <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)
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#6 {main}</pre>
rightwards double arrow          <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)
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#6 {main}</pre>                                                                                                      ...(ii)
From (i),   left parenthesis square root of straight a minus square root of straight b right parenthesis end root space left parenthesis square root of straight a plus square root of straight b right parenthesis space equals space 12 space rightwards double arrow space space 2 left parenthesis square root of straight a plus square root of straight b right parenthesis space equals space 12 space rightwards double arrow space space square root of straight a plus square root of straight b space equals space 6                      ...(iii)
Adding (ii) and (iii), we get     2 square root of straight a space equals space 8 comma space space space square root of straight a space equals space 4 space space space rightwards double arrow space space space straight a space equals space 16
Subtracting (ii) from (iii), we get 2 square root of straight b space equals space 4 space rightwards double arrow space space square root of straight b space equals space 2 space rightwards double arrow space straight b space equals space 4

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