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 Question from Jackie, a student: A specific brand of bike comes in two frames, for males or females. Each frame comes in a choice of two colors, red and blue, and with a choice of three seats, soft, medium, and hard. a) Use the counting principle to determine the number of different arrangements of bicycles that are possible. b) Construct a tree diagram illustrating all the different arrangements of bicycles that are possible. c) List the sample space. I'm really not sure where to start on this. My textbook doesn't go into detail. Thank you for any help you can give me.

Hi Jackie,

I am going to point you to a question we received a while ago that was

For lunch, students can choose juice or milk; hamburger or pizza; and a banana or apple. How many lunch combinations can the students make?

I used a tree diagram to solve this problem and I think that looking at it will help you with b) part of your problem but let's look at the counting principle first.

In the lunch problem the students have 2 choices for a drink, 2 choices for a main course and 2choices for desert. As the student goes through the line she first chooses one of two drinks. Regardless of the drink she chooses she can choose one of two main courses. Thus at this point she has chosen one of 2 2 = 4 possibilities. Now, regardless of what is on her tray already she can choose one of two desserts. Thus in total there are 2 2 2 = 8 possible lunch combinations.

If you now look at the tree diagram you will see all 8 possible lunch combinations. In the notation I used there they are {J, H, B}, {J, H, A}, {J, P, B}, and so on. This is the sample space, the list of all possible lunch combinations.

I hope this helps,
Penny

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