Question

Ten females and eight males are interviewed for five positions, if all are found to be acceptable for any position, in how many ways could the following combinations be selected?

a) Four females and four males
b) Five females and two males

Answer 1

To find the number of ways to select specific combinations from a group of males and females, we calculate the combinations for each gender separately and multiply them. For four females and four males, there are 14,700 ways; for five females and two males, there are 7,056 ways. The calculations are based on the hypergeometric distribution.

Step-by-step explanation:

The question relates to combinatorics, specifically to the combination of individuals into groups without regard to the order within the groups. We are given a pool of 10 females and 8 males and are asked to calculate the number of ways to select:

1. Four females and four males.
2. Five females and two males.

For (a), we calculate as follows:
Combinations of females = 10 choose 4 = 10! / (4!(10 - 4)!) = 210
Combinations of males = 8 choose 4 = 8! / (4!(8 - 4)!) = 70
Total combinations = 210 * 70 = 14,700

For (b), we calculate as follows:
Combinations of females = 10 choose 5 = 10! / (5!(10 - 5)!) = 252
Combinations of males = 8 choose 2 = 8! / (2!(8 - 2)!) = 28
Total combinations = 252 * 28 = 7,056

The method used involves the hypergeometric distribution, which is applicable when we are dealing with a finite population divided into two groups and selecting without replacement.