Counting Methods Topic 1: Introduction to the Fundamental Counting Principle

I can represent and solve counting problems, using a graphic organizer. I can generalize, using inductive reasoning, the fundamental counting principle. I can solve a contextual counting problem, using the fundamental counting principle, and explain the reasoning.

Explore… American Eagle Outfitters is a popular clothing store that carries both men’s and women’s fashions. Suppose a customer purchases the five colour-coordinated items shown below. ShortsT-shirts Navy (N)White (W) Black (B)Red (R) Grey (G) 1. List the outfits possible consisting of a pair of shorts and a t-shirt.

Explore… 2. A tree-diagram is another way to list all possible outcomes. Fill in the blanks in the tree-diagram shown below. The first branch is completed for you. 3. Explain how you could have numerically calculated the number of outfits, without actually listing the outfits or representing them as a tree-diagram.

Explore… 4. The customer has a choice of purchasing another colour coordinated t-shirt or pair of shorts. Make a prediction about which purchase will give him or her the greatest number of outfits. 5. Test your prediction using an organized list or a tree diagram.

You should see… 1. NW, NR, BW, BR, GW, GR 2. Tree Diagram (right) 3. Multiplying the number of short possibilities (3) by the number of t-shirt possibilities (2) will give If they bought an additional pair of shorts, they would have 4 x 2 = 8 possible outfits. If they both an additional t-shirt, they would have 3 x 3 = 9 possible outfits. 5. If you test this prediction, you will see that it is true.

Information Sample space is an organized listing of all possible outcomes from an experiment. Two common ways to organize a sample space are to use an outcome table or a tree diagram. An outcome table is a table that lists the sample space in an organized way. In a tree diagram, each branch represents a different outcome of the sample space.

Information The fundamental counting principle says that if there are a selections of one item, b selections of a second item, and c selections of a third item, then the number of arrangements of all three items is a × b × c. This can be extended for any number of items.

Example 1 The menu in an oriental restaurant lists 2 meat dishes, 2 vegetable dishes and 3 rice dishes. How many meals are possible consisting of a meat, a vegetable and a rice dish? a) Use a tree diagram to represent the sample space. b) Use the fundamental counting principle to determine how many meals are possible. Using outcome tables and tree diagrams to represent sample space.

Example 1: Solution a) There are 2 meat dishes (M1, M2) There are 2 vegetable dishes (V1, V2) There are 3 rice dishes (R1, R2, R3) M1 V1 R1 V2 R2 R3 R1 R2 R3 M1, V1, R1 M1, V1, R2 M1, V1, R3 M1, V2, R1 M1, V2, R2 M1, V2, R3 M2 V1 R1 V2 R2 R3 R1 R2 R3 M2, V1, R1 M2, V1, R2 M2, V1, R3 M2, V2, R1 M2, V2, R2 M2, V2, R3 There are 12 meal combinations.

Example 1: Solution b) Using the fundamental counting principal, 2 × 2 × 3 = 12 Again, there are 12 meal combinations.

Note If only the number of outcomes is needed, use the fundamental counting principle.

Example 2 A combination lock has 40 numbers. To open the lock, rotate clockwise to the first number, counter-clockwise past the first number to the second number and then clockwise to the third number. a) i. How many lock combinations are possible, if any number may be repeated? ii. If it takes 5 seconds to dial a complete combination, how many hours would it take to dial all possible combinations? ____ x ____ x ____ = _______ There are 3 spaces to fill, as the combination has 3 numbers. Since repetitions can be used, there are 40 possibilities for every space x 5 = seconds ÷ 60 = minutes ÷ hours Using FCP to solve a counting problem

Example 2 b) How many combinations are possible, if the middle digit is 12? ____ x ____ x ____ = _______ Since the middle digit has to be a 12, there is only one possibility for this space. 1

Example 2 c) How many combinations are possible if the first number is 28 and the last number is 47? ____ x ____ x ____ = _______ 1 Since the first digit has to be a 28, and the last number has to be a 47, there is only one possibility for each of these spaces. 40

Example 2 d) How many combinations are possible, if the numbers are 12, 28 and 47, but not necessarily in that order? ____ x ____ x ____ = _______ 3 Since there are only 3 possibilities for the first space we put in a 3. 2 One of the numbers has been used, so we have 2 possibilities left for space number of the numbers have been used so there is only 1 possibility left for the 3 rd space.

Example 3 a) A country’s postal code consists of six characters. The characteristics in the odd positions are upper-case letters, while the characters in the even positions are digits (0 to 9). How many postal codes are possible in this country? ____ x ____ x ____ x ____ x ____ x ____ = _____________ Spaces 1, 3, and 5 must be filled with upper case letters. Spaces 2, 4, and 6 must be filled with digits (0 to 9). Using FCP to solve a counting problem

Example 3 b) Canadian postal codes are similar, except the letters D, F, I, O, and U can never appear. (This is because they might be mistaken for the letters E or V or the numbers 0 or 1). How many postal codes are possible in Canada? ____ x ____ x ____ x ____ x ____ x ____ = _____________ Spaces 1, 3, and 5 must be filled with upper case letters (not D, F, I, O, or U).. Spaces 2, 4, and 6 must be filled with digits (0 to 9). Using FCP to solve a counting problem

Example 3 c) How many less postal codes are there in Canada compared to the country in part a)? =other  Canada =  = Using FCP to solve a counting problem

Need to Know The fundamental counting principle says that if there are a selections of one item, b selections of a second item, and c selections of a third item, then the number of arrangements of all three items is a × b × c. The fundamental counting principle applies when tasks are related by the word AND. When using the fundamental counting principle, address the restrictions first.

Need to Know Organized lists, outcome tables, and tree diagrams can also be used to solve counting problems when you want to list all possible outcomes. You’re ready! Try the homework from this section.