Question

Of a company's personnel, 7 people work in design, 14 in manufacturing, 4 in testing, 5 in sales, 2 in accounting, and 3 in marketing. A committee of 6 people is to be formed to meet with upper management.

a)How many committees with 1 member from each department are possible?
b)How many committees with exactly 2 members from manufacturing are possible?
c)How many committees with no representative from accounting and exactly 1 representative from marketing are possible?
d)How many committees with at least 2 representatives from manufacturing are possible?

Answer 1

a) There are 8,400 possible committees with 1 member from each department. b) There are 50,040 possible committees with exactly 2 members from manufacturing. c) There are 17,640 possible committees with no representative from accounting and exactly 1 representative from marketing. d) There are 5,880 possible committees with at least 2 representatives from manufacturing.

Step-by-step explanation:

a) To form a committee with 1 member from each department, we need to select one person from each department. There are 7 choices for the design department, 14 choices for the manufacturing department, 4 choices for the testing department, 5 choices for the sales department, 2 choices for the accounting department, and 3 choices for the marketing department. Therefore, the total number of possible committees is 7 * 14 * 4 * 5 * 2 * 3 = 8,400.

b) To form a committee with exactly 2 members from manufacturing, we need to select 2 people from the manufacturing department and 1 person from each of the other departments. There are C(14,2) = 91 ways to choose 2 people from the manufacturing department. There are 7 choices for the design department, 4 choices for the testing department, 5 choices for the sales department, 2 choices for the accounting department, and 3 choices for the marketing department. Therefore, the total number of possible committees is 91 * 7 * 4 * 5 * 2 * 3 = 50,040.

c) To form a committee with no representative from accounting and exactly 1 representative from marketing, we need to select 1 person from the marketing department and 1 person from each of the other departments (except accounting). There are 3 choices for the marketing department. There are 7 choices for the design department, 14 choices for the manufacturing department, 4 choices for the testing department, 5 choices for the sales department, and 3 choices for the marketing department. Therefore, the total number of possible committees is 3 * 7 * 14 * 4 * 5 * 3 = 17,640.

d) To form a committee with at least 2 representatives from manufacturing, we can calculate it by subtracting the number of committees with 0 or 1 representative from manufacturing from the total number of committees. The total number of committees is 8,400. The number of committees with 0 representative from manufacturing is 7 * 4 * 5 * 2 * 3 = 840. The number of committees with 1 representative from manufacturing is 14 * 4 * 5 * 2 * 3 = 1,680. Therefore, the number of committees with at least 2 representatives from manufacturing is 8,400 - 840 - 1,680 = 5,880.