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) To determine the number of different arrangements of bicycles that are possible, we can apply the counting principle. Each choice (frame, color, seat) can be treated as a separate step, and the total number of arrangements is found by multiplying the number of options at each step.

Step 1: Choosing the frame (male or female) - 2 options.
Step 2: Choosing the color (red or blue) - 2 options.
Step 3: Choosing the seat (soft, medium, or hard) - 3 options.

Using the counting principle, we multiply the number of options at each step: 2 x 2 x 3 = 12.

Therefore, there are 12 different arrangements of bicycles that are possible.

b) The tree diagram can be constructed by visually representing each step and branching out to show the available options at each step. Here is a representation of the tree diagram:

Frame
/ \
M F
/ \ / \
R B R B
/ \ / \
S M,S H, S
/ \ \
red blue red blue
/ | |
\---red-red-red
\-red-red-blue
\-red-blue-red
\-red-blue-blue
\-blue-red-red
\-blue-red-blue
\-blue-blue-red
\-blue-blue-blue

c) The sample space refers to the complete set of all possible outcomes. In this case, since a specific brand of bike comes in two frames (male or female), two colors (red or blue), and three seats (soft, medium, or hard), the sample space can be listed as follows:

1. Male frame, red color, soft seat
2. Male frame, red color, medium seat
3. Male frame, red color, hard seat
4. Male frame, blue color, soft seat
5. Male frame, blue color, medium seat
6. Male frame, blue color, hard seat
7. Female frame, red color, soft seat
8. Female frame, red color, medium seat
9. Female frame, red color, hard seat
10. Female frame, blue color, soft seat
11. Female frame, blue color, medium seat
12. Female frame, blue color, hard seat

So, the sample space consists of these 12 different arrangements of bicycles.