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, we need to multiply the number of options for each category.

1. Gender: There are 2 frames - one for males and one for females.
2. Color: There are 2 color options - red and blue.
3. Seat: There are 3 seat options - soft, medium, and hard.

Using the counting principle, we multiply the number of options for each category:
Total number of arrangements = 2 (gender) × 2 (color) × 3 (seat) = 12.

So, there are a total of 12 different arrangements of bicycles possible.

b) Here is a tree diagram illustrating all the different arrangements:

Gender
/ \
Male Female
/ \ / \
Red Blue Red Blue
/ | \ / | \ / | \ / | \
S M H S M H S M H S M H

c) The sample space, or the complete list of all possible outcomes, can be listed as follows:
1. Male, Red frame, Soft seat
2. Male, Red frame, Medium seat
3. Male, Red frame, Hard seat
4. Male, Blue frame, Soft seat
5. Male, Blue frame, Medium seat
6. Male, Blue frame, Hard seat
7. Female, Red frame, Soft seat
8. Female, Red frame, Medium seat
9. Female, Red frame, Hard seat
10. Female, Blue frame, Soft seat
11. Female, Blue frame, Medium seat
12. Female, Blue frame, Hard seat