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:

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  =\[{}^{10} C_2 \times^{20} C_3 = \frac{10}{2} \times \frac{9}{1} \times \frac{20}{3} \times \frac{19}{2} \times \frac{18}{1} = 51300\]

If a particular student is excluded, it means that 3 students are selected out of the remaining 19 students.
Required number of ways =\[{}^{19} C_3 \times^{10} C_2 = \frac{19}{3} \times \frac{18}{2} \times \frac{17}{1} \times \frac{10}{2} \times \frac{9}{1} = 43605\]