Question:
A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?
Solution:
A committee of 3 people is to be constituted from a group of 2 men and 3 women.
$\therefore$ Number of ways $={ }^{2} C_{0} \times{ }^{3} C_{3}+{ }^{2} C_{1} \times{ }^{3} C_{2}+{ }^{2} C_{2} \times{ }^{3} C_{1}=1+2 \times 3+3 \times 1=10$
Number of committees consisting of 1 man and 2 women $={ }^{2} C_{1} \times{ }^{3} C_{2}=6$