Question:
A committee of three 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:
Total number of persons = 2 + 3 = 5
Now, committee consist of 3 persons.
Therefore, total number of ways $={ }^{5} \mathrm{C}_{3}$ $=\frac{5 !}{3 ! \times(5-3) !}$ $=5 \times 2=10$
Now,
When 1 man is selected, total ways $={ }^{2} \mathrm{C}_{1}$
When 2 women are selected, total ways $={ }^{3} \mathrm{C}_{2}$
Total number of ways when 1 man and 2 women are selected $={ }^{2} \mathrm{C}_{1} \times{ }^{3} \mathrm{C}_{2}=2 \times 3=6$
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