Final answer:
The probability of a randomly chosen 3-member committee including exactly 2 junior members is determined by calculating the combinations of selecting 2 juniors and 1 senior divided by the total combinations of picking any 3 members from the entire club.
Step-by-step explanation:
The question involves calculating the probability that a randomly chosen 3-member committee from a club with 9 junior members and 14 senior members will include exactly 2 junior members. To solve this, we use combinatorics. The total number of ways to choose a 3-member committee from the entire group is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of members and k is the number of members to choose.
To find the probability of including exactly 2 junior members, we must calculate the number of ways to choose 2 juniors out of 9 and 1 senior out of 14, and then divide by the total number of ways to select a committee of 3 from the entire group. The calculation is as follows:
Number of ways to choose 2 juniors and 1 senior = C(9, 2) * C(14, 1)
The number of ways to choose 3 members from the group = C(23, 3)
Probability = (C(9, 2) * C(14, 1)) / C(23, 3)
The combinatorial expressions represent the number of combinations of selecting a certain number of members from the junior or senior groups without regard to the order.