**Question:**

Find the number of ways in which a committee of 2 teachers and 3 students can be formed out of 10 teachers and 20 students. In how many of these committees

(i) a particular teacher is included?

(ii) a particular student is included?

(iii) a particular student is excluded?

**Solution:**

Since a committee is to be formed of 2 teachers and 3 students

(i) When a particular teacher is included

No. of ways in which committee can be formed $={ }^{9} \mathrm{C}_{1} \times{ }^{20} \mathrm{C}_{3}$

= 9720 ways

(ii) A particular student is included

Since a particular student is always selected so ways of selecting 2 teachers and 2 students from 10 and 19 respect. is $={ }^{10} \mathrm{C}_{2} \times{ }^{19} \mathrm{C}_{2}$ ways

= 7695 ways

(iii) A particular student is excluded

Since 1 particular student is excluded so, ways of selecting 2 teachers and 3 students from 10 and 19 respt. is $={ }^{10} \mathrm{C}_{2} \times{ }^{19} \mathrm{C}_{3}$ ways

= 43605 ways