Final answer:
The student's problems involve calculating the probability of selecting a committee with members from specific groups and the total number of committees that can be formed from a given set of men and women. These scenarios are modeled using combinatorial probability with hypergeometric distribution and combination formulas.
Step-by-step explanation:
The problem presented is a classic example of a combinatorial probability question in mathematics, where we calculate the chance of selecting a subset from a larger set without replacement. This is described as a hypergeometric distribution problem because members are chosen from two distinct groups.
Probability of All Democrats Being Chosen
To determine the probability that all of the five chosen members are Democrats, we use the following formula:
P(All Democrats) = (Number of ways to choose 5 Democrats from 8) / (Total number of ways to choose 5 members from 18 (8 Democrats + 10 Republicans))
The calculation is thus C(8, 5) / C(18, 5), where C(n, k) represents the combination formula n! / (k! (n-k)!).
Committees of 5 Women and 5 Men
For the second question, the number of different committees that can be formed is the product of the number of ways to choose 5 women out of 7 and 5 men out of 9:
Total committees = C(7, 5) * C(9, 5)