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Permutations and Combinations

Short Answer Type

41. Find L.C.M. and H.C.F. of 2 x 2!, 4! and 3 x 5!

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42.

Show that :

$space space space space space space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times........... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$

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43. Prove that n! + (n + 1)! = [n + 2] x n !

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44. Show that 2¹⁶ divides 32!. Also, find the largest value of n for which 32! is divisible by 2ⁿ. Or What is the largest power of 2 contained in 32!

32! = 32 x 31 x 30 x 29 x ......... x 3 x 2 x 1

The product contains 32 factors in all of which 16 are even factors. Each even number is divisible by 2.

∴ 32! is divisible by 2¹⁶.

Also, the powers of 2 contained in 32!

2⁵ from 32, 2¹ from 30, 2² from 28, 2¹ from 26, 2³ from 24, 2¹ from 22, 2² from 20, 2¹ from 18,

2⁴ from 16, 2¹ from 14, 2² from 12, 2¹ from 10, 2³ from 8, 2¹ from 6, 2² from 4 and 2¹ from 2.

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$rightwards double arrow$ $2 to the power of 31$ divides 32!

Hence, the largest value of n for which $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ divides 32! is n = 31.