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.

a) What we can do here is multiply the choices out:

(3 seats) x (2 colors) x (2 frames) = 12 combinations

b) We can make a standard tree diagram with the top node being the choice in frames. Expanding that, at each node, add a choice for both colors, and then at each resulting node (there should be 4 at this point), add three choices each for the seats. You should end up with 12 nodes.

c) The sample space lists each outcome within a set of braces. So you have {(Male, blue, hard seat); (female, blue, hard seat);...}

thanks Perez, You have been a great help.

a) To determine the number of different arrangements of bicycles, we need to multiply the number of options for each component together.

In this case, we have:
- 2 frames (for males or females)
- 2 colors (red or blue)
- 3 seats (soft, medium, or hard)

We can use the counting principle, also known as the multiplication principle, which states that if there are m ways to do one thing, and n ways to do another, then there are m * n ways to do both.

Therefore, the number of different arrangements of bicycles would be:
2 (frames) * 2 (colors) * 3 (seats) = 12 different arrangements.

b) To construct a tree diagram illustrating all the different arrangements of bicycles, we can start by creating branches for each component.

First, we start with the frame:
- One branch for the male frame
- One branch for the female frame

Next, for each frame option, we add branches for the color options:
- For the male frame branch, we add branches for red and blue
- For the female frame branch, we also add branches for red and blue

Finally, for each frame and color option, we add branches for the seat options:
- For each color option under the male frame branch, we add branches for soft, medium, and hard seats
- Likewise, for each color option under the female frame branch, we add branches for all three seat options as well

The resulting tree diagram would have four branches at the first level (male frame, female frame, red color, blue color) and three branches at the second level (soft seat, medium seat, hard seat) under each of these four options.

c) The sample space is the list of all possible outcomes or arrangements. In this case, the sample space would consist of all the different arrangements of bicycles that we determined in part (a).

Given the multiplication principle, the sample space would consist of 12 different arrangements of bicycles. Each arrangement would represent a unique combination of frame, color, and seat options.