Permutation and Combination

Question

A work crew for a building project is to be made up of 2 craftsmen and 4 laborers selected from a total of 5 craftsmen and 6 laborers.

a. How many different combinations are possible?

b. The brother of one of the craftsmen is a laborer. If the crew is selected at random, what is the probability that both brothers will be selected?

c. What is the probability that neither brother will be selected?

Answer 1

a) 2 of 5 craftsmen, 4 of 6 laborers

C(5,2)C(6,4) = C(5,2)C(6,2) = 10 * 15 = 150

b) We can use the result in a, and say the number of combinations including each brother means 1 of 4 other craftsmen and 3 of 5 laborers =

C(4,1)*C(5,3) = C(4,1)*C(5,2) = 4 * 10 = 40

Thus, the probability of selecting both brothers is 40/150 = 4/15

An alternate way:

Since we select 2 of 5 craftsmen, one brother has a 2/5 chance of being selected. Since we select 4 of 6 laborers, the other brother has a 4/6 = 2/3 chance of being selected.

Thus, the chance of both being selected is 2/5 * 2/3 = 4/15 The answers match.

c) The probability that neither brother will be selected.

Using combinations, this means selecting 2 of 4 craftsmen and 4 of 5 laborers.

C(4,2)*C(5,4) C(4,2)*C(5,1) = 6 * 5 = 30

Thus, the probability of selecting neither brother is 30/150 = 1/5

The other way.

The probability of not selecting the brother as a craftsman is 1 - 2/5 = 3/5. The probability of not selecting the brother as a laborer is 1 - 4/6 = 1 - 2/3 = 1/3

3/5 * 1/3 = 1/5. Again, the answers match.