Final answer:
The number of possible committees that can be formed are 2,160 for one person from each class, 27,405 for any mixture, and 10,098 for exactly two seniors.
Step-by-step explanation:
The question revolves around the concept of combinations and the different ways to form a committee of four from a group of students in various classes, with certain restrictions. Let's tackle each part of the question:
(a) One Person from Each Class
To select one senior, one junior, one sophomore, and one freshman for the committee, you would calculate the number of combinations possible for each class and then multiply them together since these are independent choices. This is calculated as:
- 12 seniors choose 1: 12C1 = 12 ways
- 9 juniors choose 1: 9C1 = 9 ways
- 5 sophomores choose 1: 5C1 = 5 ways
- 4 freshmen choose 1: 4C1 = 4 ways
Multiplying these together: 12 x 9 x 5 x 4 = 2,160 possible committees.
(b) Any Mixture of the Classes
- The total number of students is 12 + 9 + 5 + 4 = 30 students. To choose any combination of four students regardless of class, we use the combination formula 30C4, which gives the number of possible committees:27,405.
(c) Exactly Two Seniors
To have a committee with exactly two seniors and two students from the remaining classes, you select two seniors from the 12 and two students from the remaining 18 (9 juniors, 5 sophomores, and 4 freshmen) students. Thus, you calculate 12C2 to choose the seniors and 18C2 for the remaining students, then multiply:
- 12 seniors choose 2: 12C2 = 66 ways
- 18 non-seniors choose 2: 18C2 = 153 ways
Multiplying these together: 66 x 153 = 10,098 possible committees.