Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Permutations and Combinations

Short Answer Type

61.

Find the value of:

$fraction numerator left parenthesis straight n plus 3 right parenthesis factorial over denominator left parenthesis straight n plus 1 right parenthesis factorial end fraction$

92 Views

62.

Convert the following to factorial notation;

(a) $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ (b) $5 times 6 times 7 times 8 times 9 times 10 times 11 times 12$

91 Views

63.

Show that (n - 1)! + n! = [n+1] x (n - 1)!

100 Views

64.

Find the largest value of n so that 2ⁿ divides 16!

105 Views

65.

Prove that 33! is divisible by $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ What is the largest value of n so that 33! is divisible by $space space space 2 to the power of straight n$ ?

101 Views

66.

Find the value of n such that:

$straight P presuperscript straight n subscript 5 space equals space 42 straight P presuperscript straight n subscript 3 comma space straight n greater than 4$

84 Views

67.

Find the value of n such that:

$space space fraction numerator straight P presuperscript straight n subscript 4 over denominator straight P presuperscript straight n minus 1 end presuperscript subscript 4 end fraction equals 5 over 3 comma space straight n greater than 4$

86 Views

68.

Find r, if

$space straight P presuperscript 5 subscript straight r space equals space 2. straight P presuperscript 6 subscript straight r minus 1 end subscript$

121 Views

69.

Find r, if

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

95 Views

70.
Find r, if

$space space space space 5. space straight P presuperscript 4 subscript straight r space equals space 6. space straight P presuperscript 5 subscript straight r minus 1 end subscript$

   $5 times space straight P presuperscript 4 subscript straight r space equals space 6 times space straight P presuperscript 5 subscript straight r minus 1 end subscript$

$rightwards double arrow$ $5 times space fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space 6 times space fraction numerator 5 factorial over denominator left parenthesis 5 minus straight r plus 1 right parenthesis factorial end fraction$

$rightwards double arrow$    $5 times fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space 6 times space fraction numerator 5 factorial over denominator left parenthesis 6 minus straight r right parenthesis factorial end fraction$

$rightwards double arrow$ $5 times fraction numerator 4 factorial over denominator left parenthesis 4 minus straight r right parenthesis factorial end fraction space equals space fraction numerator 6 cross times 5 cross times 4 factorial over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis space left parenthesis 4 minus straight r right parenthesis factorial end fraction$

$rightwards double arrow$ $5 equals fraction numerator 30 over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis end fraction$

$rightwards double arrow$    $1 equals fraction numerator 6 over denominator left parenthesis 6 minus straight r right parenthesis space left parenthesis 5 minus straight r right parenthesis end fraction$

$rightwards double arrow$    $straight r squared minus 11 straight r plus 30 space equals space 6$

$rightwards double arrow$ $straight r squared minus 11 straight r plus 24 equals 0$

$rightwards double arrow$ (r - 8) (r - 3) = 0

$rightwards double arrow$ r = 8, r = 3

But, r(=8) > n (= 4) [Not valid]

Hence, r = 3