Permutations And Combinations

Question
CBSEENMA11014212

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? 

Solution

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: <b>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</b><br /><br />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<br />#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)
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#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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