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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?

Number of ways in which he can enter = 2
$rightwards double arrow$ m = 2
Number of ways in which he can leave the room = 3
$rightwards double arrow$ n = 3
By fundamental principle of counting, the number of ways in which he can enter the room from one side and get out from the outer side = m x n = 2 x 3 = 6

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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

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250.

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

8 . ⁶C₄ . ⁷C₄
6 . 7 . ⁸C₄
6 . 8 . ⁷C₄
6 . 8 . ⁷C₄

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