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Question

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:
(i) a particular professor is included.
(ii) a particular student is included.
(iii) a particular student is excluded.

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Solution

Clearly, 2 professors and 3 students are selected out of 10 professors and 20 students, respectively.
Required number of ways = 10C2×20C3 = 102×91×203×192×181 = 51300

(i) If a particular professor is included, it means that 1 professor is selected out of the remaining 9 professors.
Required number of ways = 20C3×9C1 = 203×192×181×91 = 10260

(ii) If a particular student is included, it means that 2 students are selected out of the remaining 19 students.
Required number of ways = 19C2×10C2 = 192×181×102×91 = 7695

(iii) If a particular student is excluded, it means that 3 students are selected out of the remaining 19 students.
Required number of ways = 19C3×10C2 = 193×182×171×102×91 = 43605

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