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.

Answer 1

Clearly, 2 professors and 3 students are selected out of 10 professors and 20 students, respectively.
Required number of ways = ${}^{10}{C}_2{\times }^{20}{C}_3=\frac{10}{2}\times \frac{9}{1}\times \frac{20}{3}\times \frac{19}{2}\times \frac{18}{1}=51300$

(i) If a particular professor is included, it means that 1 professor is selected out of the remaining 9 professors.
Required number of ways = ${}^{20}{C}_3{\times }^9{C}_1=\frac{20}{3}\times \frac{19}{2}\times \frac{18}{1}\times \frac{9}{1}=10260$

(ii) If a particular student is included, it means that 2 students are selected out of the remaining 19 students.
Required number of ways = ${}^{19}{C}_2{\times }^{10}{C}_2=\frac{19}{2}\times \frac{18}{1}\times \frac{10}{2}\times \frac{9}{1}=7695$

(iii) If a particular student is excluded, it means that 3 students are selected out of the remaining 19 students.
Required number of ways = ${}^{19}{C}_3{\times }^{10}{C}_2=\frac{19}{3}\times \frac{18}{2}\times \frac{17}{1}\times \frac{10}{2}\times \frac{9}{1}=43605$