From a group of 10 men and 8 women, the number of ways of forming a committee of 8 members with not more…
From a group of 10 men and 8 women, the number of ways of forming a committee of 8 members with not more than 5 men and not less than 5 women is
8061
8060
20997
20952
Solution
From group of 10 men and 8 women committee of 8 members with not more than 5 men and not less than 5 women is formed in following ways.
Case I. If 3 men and 5 men we selected number of ways $={ }^{10} C_3 \times{ }^8 C_5$
$
=\frac{10 \times 9 \times 8}{3 \times 2} \times \frac{8 \times 7 \times 6 \times 5 \times 4}{5 \times 4 \times 3 \times 2 \times 1}=6720
$
Case II. If 2 men and 6 women are selected Number of ways $={ }^{10} C_2 \times{ }^8 C_6$
$
=\frac{10 \times 9}{2} \times \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3}{6 \times 5 \times 4 \times 3 \times 2}=1260
$
Case III. If 1 man and 7 women are selected number of ways $={ }^{10} C_1 \times{ }^8 C_7=10 \times 8=80$
Case IV. If 0 men and 8 women are selected number of ways $={ }^{10} C_0 \times{ }^8 C_8=1$ total number of required ways
$
=6720+1260+80+1=8061
$