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Permutations and Combinations

Long Answer Type

171. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has:
(i) no girl (ii) at least one boy and one girl (iii) at least 3 girls.

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Short Answer Type

172.

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

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Long Answer Type

173.

What is the number of ways of choosing 4 cards from a pack of 52 cards? In how many of these :

(i) four cards are of the same suit?

(ii) four cards belongs to different suit?

(iii) are face cards?

(iv) two are red cards and two are black cards?

(v) cards are of the same colour?

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Short Answer Type

174.

How many chords can be drawn through 21 points on a circle?

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175.

How many straight lines can be formed with 7 points, 3 of which are colinear.

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176. Find the number of triangles formed by 9 points lying on the circumference of a circle.

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177. Find the number of triangles formed by 12 points, 5 of which are collinear.

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178.

If $straight C presuperscript straight n minus 1 end presuperscript subscript straight r colon straight C presuperscript straight n subscript straight r colon straight C presuperscript straight n plus 1 end presuperscript subscript straight r space equals space 6 colon 9 colon 13 comma$ Find n and r.

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179. In an examination, a question paper consists of 12 questions, divided into two parts i.e., Part I and Part II containing 5 and 7 questions respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can the student select his questions?

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180. From a class of 25 students, 10 are to be chosen for an excursion party. There are three students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Total number of students = 25
The number of students to go for excursion = 10
Options are:
All 3 of them join the party and 7 others join the party from remaining 22 students.

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

All there do not join the party and 10 join the party from remaining 22 students.

$rightwards double arrow$ Number of selections = $straight C presuperscript 3 subscript 0 cross times straight C presuperscript 22 subscript 10 space equals space 1 space cross times space straight C presuperscript 22 subscript 10 equals space straight C presuperscript 22 subscript 10$ ...(ii)
Hence, the total number of ways of selecting 10 out of 25 for an excursion party

= $space space straight C presuperscript 22 subscript 7 plus straight C presuperscript 22 subscript 10 space equals space fraction numerator 22 factorial over denominator 15 factorial space space 7 factorial end fraction. space fraction numerator 22 factorial over denominator 12 factorial space 10 factorial end fraction$

= 170544 + 646646 = 817190.

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