1

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.

(i) a particular professor is included.

(ii) a particular student is included.

(iii) a particular student is excluded.

Open in App

Solution

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 = ${}^{19}{C}_{3}{\times}^{10}{C}_{2}=\frac{19}{3}\times \frac{18}{2}\times \frac{17}{1}\times \frac{10}{2}\times \frac{9}{1}=43605$

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 = ${}^{19}{C}_{3}{\times}^{10}{C}_{2}=\frac{19}{3}\times \frac{18}{2}\times \frac{17}{1}\times \frac{10}{2}\times \frac{9}{1}=43605$

0

View More

Join BYJU'S Learning Program

Join BYJU'S Learning Program