Question

1. (a) Suppose that, from 12 students, we want to take a committee of students. How many committees consisting of 4 students can be selected? How many committees consisting of 6 students can be selected?

(b) Suppose we want to select 3 out of 12 students and give them the first and second and third places. Determine the number of possible selections?

Answer 1

Answer a.

Committee of 4 students has to be formed out of 12 students:

Number of ways in which 4 students can be choose from 12 students = ¹²C₄
Number of ways in which 4 students can be choose from 12 students = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1)
Number of ways in which 4 students can be choose from 12 students = 11 * 5 * 9
Number of ways in which 4 students can be choose from 12 students = 495

Committee of 6 students has to be formed out of 12 students:

Number of ways in which 6 students can be choose from 12 students = ¹²C₆
Number of ways in which 6 students can be choose from 12 students = (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1)
Number of ways in which 6 students can be choose from 12 students = 924

Answer b.

1^st Place:

Number of ways in which 1^st place is allotted out of 12 students = 12
Number of ways in which 2^nd place is allotted out of remaining 11 students = 11
Number of ways in which 3^rd place is allotted out of remaining 10 students = 10

Total Ways in which 1^st, 2^nd and 3^rd place can be allotted = 12 * 11 * 10
Total Ways in which 1^st, 2^nd and 3^rd place can be allotted = 1320