LEQ: What Is the Difference Between a Permutation and a Combination?

Statistics 3.4 Counting Principles

LEQ: What is the difference between a permutation and a combination?

Procedure:

1. The Fundamental Counting Principle:

a. Definition 1: ______: If one event can occur in m ways, and a second event can occur in n ways, the number of ways the two events can occur in sequence is m x n. This rule can be extended for any number of events occurring in sequence.

b. Examples 1 & 2: Using the fundamental counting principle:

1. You are purchasing a new car. Use the following manufacturers, car sizes, and colors, how many different ways can you select one manufacturer, one car size, and one color?

Manufacturer: Ford, GM, Chrysler Car Size: small, medium Color: white, red, black, green

2. How many license plates can you make if a license plate consists of:

a. Six letters each of which can be repeated?

b. Six letter each of which cannot be repeated? 2. Permutations:

a. Definition 2: A ______is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!.

b. Example 3: Finding the number of permutations of n objects:

3. The starting lineup for a baseball team consists of nine players. How many different batting orders are possible using the starting lineup?

c. Definition 3: The number of permutations of n distinct objects taken r at a time is

d. Examples 4 & 5: Finding :

4. Find the number of ways of forming three-digit codes in which no digit is repeated?

5. Forty-three race cars started the 2004 Daytona 500. How many ways can cars finish first, second, and third?

e. Definition 4: The number of ______of n objects, where n1 are of one type, n2 are of another type, and so on is

f. Example 6: Distinguishable permutations: 6. A building contractor is planning to develop a subdivision. The subdivision is to consist of 6 one-story houses, 4 two-story houses, and 2 slit-level houses. In how many distinguishable ways can the houses be arranged?

3. Combinations:

a. Definition 5: A ______is a selection of r objects from a group of n objects without regard to order and is denoted by . The number of combinations of r objects selected from a group of n objects

b. Example 7: Finding the number of combinations:

7. A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies?

3. Applications of Counting Principles:

a. Example 8: Finding probabilities:

A word consists of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi?

b. Example 9: Finding probabilities: Find the probability of being dealt five diamonds from a standard deck of playing cards. (In poker, this is a diamond flush).

4. HW: p. 157 (2, 3 – 39 mo3)