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 in how many of these committees

1. a particular professor is included?
2. a particular student is excluded?

Answer

We can select 2 professors from 10 in ¹⁰C₂ ways and 3 students from 20 in ²⁰C₃ ways. Hence a committee of 2 professors and 3 students can be formed in

1. When 1 particular professor included the committee then we can select remaining 1 professor from remaining 9 in ⁹C₁ ways and 3 students from 20 in ²⁰C₃ ways. Hence a particular professor included then the number of committee can be formed in
   
2. When 1 particular student excluded from the committee then we can select 3 students from remaining 19 students in ¹⁹C₃ ways and 2 professors from 10 in ¹⁰C₂ ways. Hence a particular student excluded from the committee then the number of committee can be formed in