Answer:
\(1225\).
Step-by-step explanation:
The combination formula gives number of ways to choose \(r\) objects out of a collection of \(n\) total objects is:
\(\displaystyle \genfrac{(}{)}{0}{}{n}{k} = \frac{n!}{k!\, (n - k)!}\).
In this question, note that the choice of the \(3\) women is independent from the choice of the \(3\) men. Hence, start by finding the number of ways to choose women and men separately using the combination formula.
The number of ways to choose \(3\) women (\(k = 3\)) from the group of \(7\) (\(n = 7\)) is:
\(\displaystyle \genfrac{(}{)}{0}{}{7}{3} = \frac{7!}{3!\, (7 - 3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\).
In other words, there are \(35\) different ways to select the \(3\) women out of the group of \(7\) women. Similarly, there would be \(35\) different ways to choose \(3\) men out of a group of \(7\) men.
The choice of the \(3\) women does not depend on the choice of the \(3\) men. Hence, for each of the \(35\!\) ways to choose \(3\) women from the group of women, there would be \(35\) unique ways to choose \(3\) men from the group of men.
Overall, the unique number of ways to choose \(3\) women and \(3\) men would be the product of the two \(35 \times 35 = 1225\).
