Final answer:
The question involves calculating the number of ways to form a committee of seven with at most three women from a group containing 18 women and 15 men, using combinatorics and hypergeometric distribution principles.
Step-by-step explanation:
The question presented is a combinatorics problem that requires an understanding of hypergeometric distribution. Given a group of 33 students (18 women and 15 men) and the requirement to create a committee of seven with at most three women, we must consider the possible combinations of men and women that satisfy this condition while forming a committee. The combinations include 0, 1, 2, or 3 women on the committee. To calculate each of these scenarios involves using the combination formula C(n, k) = n! / (k! * (n - k)!), where 'n' is the total number of items to pick from, and 'k' is the number of items to pick.
For example, if the committee has three women, there needs to be four men. The number of ways to choose three women from the 18 available is C(18, 3), and the number of ways to choose four men from the 15 available is C(15, 4). This pattern continues for committees with two women and five men, one woman and six men, and no women and seven men. The total number of possible committees is the sum of all these individual scenarios.
Through this approach, students can learn how to apply the principles of combinatorics and probability to real-world situations, such as forming committees with specific characteristics from a larger pool of people.