stats final

Permutation vs. Cominations

Permutation: 1. picking a team captain, pitcher, and shortstop from a group 2. picking your favorite two colors, in order, from a color brochure. 3. picking first, second and third place Combination: 1. picking three team members from a group 2. picking two colors from a color brochure 3. picking three winners

Given a class of 12 girls and 10 boys. (a) In how many ways can a committee of five consisting of 3 girls and 2 boys be chosen?

(a) First note that the order of the children in the committee does not matter. From 12 girls we can choose C(12, 3) different groups of three girls. From the 10 boys we can choose C(10, 2) different groups. Thus, by the Fundamental Principle of Counting the total number of committee is C(12, 3) · C(10, 2) = (12 · 11 · 10 /3 · 2 · 1) · (10 · 9 /2 · 1) = 9900

How many ways can two slices of pizza be chosen from a plate containing one slice each of pepperoni, sausage, mushroom, and cheese pizza.

In choosing the slices of pizza, order is not important. This arrangement is a combination. Thus, we need to find C(4, 2) = 4! 2!(4−2)! = 6. So, there are six ways to choose two slices of pizza from the plate

How many five-digit zip codes can be made where all digits are unique? The possible digits are the numbers 0 through 9.

P(10, 5) = 10! (10−5)! = 30, 240 zip codes

Given a class of 12 girls and 10 boys. How many of the possible committees of five have no boys?(i.e. consists only of girls)

The number of ways to choose 5 girls from the 12 girls in the class is C(10, 0) · C(12, 5) = C(12, 5) = 792

Given a class of 12 girls and 10 boys. What is the probability that a committee of five, chosen at random from the class, consists only of girls?

The probability that a committee of five consists only of girls is C(12, 5) /C(22, 5) = 792/ 26, 334 = 0.03

Given a class of 12 girls and 10 boys. b. What is the probability that a committee of five, chosen at random from the class, consists of three girls and two boys?

The total number of committees of 5 is C(22, 5) = 26, 334. Using part (a), we find the probability that a committee of five will consist of 3 girls and 2 boys to be C(12, 3) · C(10, 2)/ C(22, 5) = 9900/ 26, 334 = 0.3759.

How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of 3 faculty members from the mathematics department and 4 from the computer science department, if there are 9 faculty members of the math department and 11 of the CS department?

There are C(9, 3) · C(11, 4) = 9! 3!(9−3)! · 11! 4!(11−4)! = 27, 720 ways.

How many ways can gold, silver, and bronze medals be awarded for a race run by 8 people?

Using the permuation formula we find P(8, 3) = 8! (8−3)! = 336 ways.