math

PRE-CLASS ASSIGNMENT - COUNTING METHODS

Read through sections 8.1 and 8.2 in your textbook. Be prepared to answer questions in class based on the problems below.

Imagine that you are picking out what to wear to school. The uniform you must wear consists of 2 shirts (navy, white) 2 pairs of pants (navy, khaki) and 2 ties (red, blue). Abbreviate them to NS and WS for the shirts, NP and KP for the pants and RT and BT for the ties.

1) Assuming that you must choose a shirt, a pair of pants and a tie, list all of the different possible outfits you can make:

2) How many outfits did you list?

This is a typical counting problem. It didn't take long because your choices were limited. If there had been 7 different shirts or 5 different pants, it would have been a lot more difficult. Ultimately, we need to come up with a better method than listing everything out.

The best way to approach this sort of problem is by using The Task Method, also known as The Multiplication Principle. You list out all of the tasks that you are asked to do in the problem, figure out how many possible options there are for each task, and then multiply them.

So go back to standing in your closet. There are 3 things you must do in order to get dressed:

3) Task 1: Pick a shirt. How many options do you have?

4) Task 2: Pick a pair of pants: How many options do you have?

5) Task 3: Pick a tie: How many options do you have?

6) Now multiply all of your answers from above to get the answer to the problem. (It should match your original answer.)

7) Does the order in which you do your tasks matter?

Here is another example: Let’s go out for ice cream. At the store, you have 3 different cones to choose from, 10 different flavors and 5 different toppings.

7) Assuming you choose one of each, what are the 3 tasks you must do in order to complete this problem? List them along with how many options you have for each task.

Task1:

Task 2:

Task 3:

8) How many different ice cream cones can you make?

Let’s say that your sister goes with you and insists on ordering everything different from you. She still has 3 tasks, but her options are different from yours.

9) How many different cones does she have to choose from?

10) How many different flavors?

11) How many different toppings?

12) So how many different ice cream cones can she make?

13) Define:

a) Permutation:

b) Combination:

c) Factorial:

14) Using your calculator, find 7!:

15) Using your calculator, find P(7,4):

16) Using your calculator, find C(8,3):

17) How would you explain to someone who was not in the course what the difference was between a Permutation and Combination?

18) Determine if the following are permutations or combinations:

a) Choosing fruit for a fruit salad.

b) Lining up books on a bookshelf.

c) Picking a captain and a co-captain for the football team.

d) Picking marbles from a jar.