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Permutations and Combinations

Short Answer Type

241. From 4 officers and 8 jawans. in how many ways can 6 be selected, so that:
(a) exactly one officer is included? (b) at least 2 officers are included?

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242.

Show that; $straight C presuperscript straight n subscript straight r space equals space straight n over straight r space straight C presuperscript straight n minus 1 end presuperscript subscript straight r minus 1 end subscript.$

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243.

If $straight C presuperscript 16 subscript straight r space equals space straight C presuperscript 16 subscript straight r minus 2 end subscript comma$ find $space straight C presuperscript straight r subscript 4.$

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244.

If $straight alpha space equals space straight C presuperscript straight m subscript 2 comma$ find the value of $straight C presuperscript straight alpha subscript 2.$

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245.

Evaluate:

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

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246.

If $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ and $straight C presuperscript straight n subscript straight r equals 120$ , find value of $straight C presuperscript 6 subscript straight r$

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247. There are 5 doors in a room, 2 on one side and 3 on the other side. A man wishes to enter the room from the side with two doors and get out from the other side. In how many ways can this be done?

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Multiple Choice Questions

248.

The number of integers greater than 6000 that can be formed, using the digits 3,5,6,7 and 8 without repetition, is

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249.
If m is the AMN of two distinct real numbers l and n (l,n>1) and G₁, G₂, and G3 are three geometric means between l and n, then $straight G subscript 1 superscript 4 space plus 2 straight G subscript 2 superscript 4 space plus space straight G subscript 3 superscript 4$ equals

4l² mn

4lm²n

4 lmn²

4 lmn²

4lm²n

Given,
m is the AM of l and n

l +n = 2m

and G₁, G₂, G₃, n are in GP
Let r be the common ratio of this GP
G1 = lr
G2 =lr²
G3= lr³
n = lr⁴

$rightwards double arrow space straight r space equals space open parentheses straight n over l close parentheses to the power of 1 divided by 4 end exponent Now comma space straight G subscript 1 superscript 4 space plus space 2 straight G subscript 2 superscript 4 space plus space straight G subscript 3 superscript 4 space equals left parenthesis l straight r right parenthesis to the power of 4 space plus space left parenthesis l straight r squared right parenthesis to the power of 4 space plus space left parenthesis l straight r cubed right parenthesis to the power of 4 space equals space straight l to the power of 4 space straight x space straight r to the power of 4 left parenthesis 1 plus 2 straight r to the power of 4 plus straight r to the power of 8 right parenthesis equals space straight l to the power of 4 space straight x space straight r to the power of 4 space left parenthesis straight r to the power of 4 space plus 1 right parenthesis squared equals space straight l to the power of 4 space straight x space straight n over straight l open parentheses fraction numerator n italic plus italic 1 over denominator l end fraction close parentheses equals space l n italic space x italic 4 m to the power of italic 2 italic space italic equals italic space italic 4 l m to the power of italic 2 n$