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

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162.

If $space space straight C presuperscript 10 subscript straight r space equals space straight C presuperscript 10 subscript 2 comma$ find the values of r.

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163.

Determine n if:

$straight C presuperscript 2 straight n end presuperscript subscript 3 colon straight C presuperscript straight n subscript 3 space equals space 12 colon 1$

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164. Determine n if:

straight C presuperscript 2 straight n end presuperscript subscript 3 space colon space straight C presuperscript straight n subscript 3 space equals space 11 colon 1

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165. In how many ways can a team of 3 boys and 3 girls be selected out of 5 boys and 4 girls?

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166. A bag contains 5 different black balls and 6 different red balls. Determine the number of ways in which 2 black and 3 red balls can be selected?

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167.

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

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168. The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

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169. How many words, with or without meaning each of 3 vowels and 2 consonants can be formed from the letters of the word ‘INVOLUTE’?

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Long Answer Type

170.
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done, when the committee consists of :

(i) exactly 3 girls? (ii) at least three girls? (iii) at most 3 girls?

Number of boys = 9
Number of girls = 4
The committee is to consist of 7

(i) The committee consists of exactly 3 girls
$rightwards double arrow$ Number of boys = 7 - 3 = 4

∴ The number of selections = $straight C presuperscript 9 subscript 4 cross times straight C presuperscript 4 subscript 3 space equals space fraction numerator 9 factorial over denominator 4 factorial space 5 factorial end fraction space cross times space fraction numerator 4 factorial over denominator 3 factorial space 1 factorial end fraction$

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

(ii) The committee consists of at least 3 girls.
$rightwards double arrow$ The committee consists of 3 girls + 4 boys or 4 girls + 3 boys
(Since, the number of girl is 4, so there is no further option)
When committee consists of 3 girls + 4 boys
Number of selections = $straight C presuperscript 4 subscript 3 cross times straight C presuperscript 9 subscript 4 space equals space 504$
When committee consists of 4 girls + 3 boys

Number of selections = $straight C presuperscript 4 subscript 4 cross times straight C presuperscript 9 subscript 3 space equals space 1 cross times fraction numerator 9 cross times 8 cross times 7 cross times 6 factorial over denominator 6 factorial space space 3 factorial end fraction space equals space 84.$

Hence, the number of selections for the committee = 504 + 84 = 588.

(iii) The committee consists of at most 3 girls.
This gives us the following options:
No girl + 7 boys

Number of selections = $space space straight C presuperscript 4 subscript 0 cross times straight C presuperscript 9 subscript 7 space equals space 1 cross times fraction numerator 9 factorial over denominator 2 factorial space space 7 factorial end fraction space equals space 36.$
Or
1 girl + 6 boys

Number of selections = $straight C presuperscript 4 subscript 1 cross times straight C presuperscript 9 subscript 6 space equals space 4 cross times fraction numerator 9 cross times 8 cross times 7 cross times 6 factorial over denominator 6 factorial space space 3 factorial end fraction space equals space 4 cross times 84 equals 336.$
Or
2 girls + 5 boys

Number of selections = $straight C presuperscript 4 subscript 2 cross times straight C presuperscript 9 subscript 5 space equals space fraction numerator 4 factorial over denominator 2 factorial space 2 factorial end fraction cross times fraction numerator 9 factorial over denominator 4 factorial space 5 factorial end fraction$

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Or
3 girls + 4 boys
Number of selections = $space space straight C presuperscript 4 subscript 3 cross times straight C presuperscript 9 subscript 4 space equals space 4 cross times fraction numerator 9 cross times 8 cross times 7 cross times 6 cross times 5 factorial over denominator 5 factorial space 4 factorial end fraction equals 504.$
Hence, the total number of selections = 36 + 336 + 756 + 504 = 1632.