I see the file you’ve uploaded contains questions about sample spaces and the counting principle. I'll go through the questions, solve them, and provide explanations. Let me know which specific problems you'd like detailed explanations for.

Sample Space Representations

1. Sandwich Shop:
   Sample space = {Ham, Turkey, Chicken}

2. Chess Club Practice Days:
   Sample space = {Tuesday, Wednesday, Thursday}

3. Chess Club Practice Days and Times:
   Sample space = {Tuesday 3 PM, Tuesday 4 PM, Tuesday 5 PM, Wednesday 3 PM, Wednesday 4 PM, Wednesday 5 PM, Thursday 3 PM, Thursday 4 PM, Thursday 5 PM}

4. Button Press Program Outputs Odd Numbers Twice:
   Odd numbers between 1 and 9 are {3, 5, 7}.
   Pressing the button twice:
   Sample space = {(3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7), (7, 3), (7, 5), (7, 7)}

5. Spinner (Red or Blue) and Die Roll (1-6):
   Sample space = {(Red, 1), (Red, 2), (Red, 3), (Red, 4), (Red, 5), (Red, 6), (Blue, 1), (Blue, 2), (Blue, 3), (Blue, 4), (Blue, 5), (Blue, 6)}

6. Trivia Team (Two Boys and One Girl):
   Let B1 and B2 represent the boys and G represent the girl. For the first and second question:
   Sample space = {(B1, B2), (B1, G), (B2, B1), (B2, G), (G, B1), (G, B2)}

Counting the Sample Space Outcomes

7. Jewelry Store Rings (Gold and Platinum with 4 Gemstones):
   Gold/Platinum options = 2, Gemstone options = 4.
   Total outcomes = $2 \times 4 = 8$.

8. Spinner (Red, Blue, Green) Spins Twice:
   Outcomes per spin = 3.
   Total outcomes = $3 \times 3 = 9$.

9. Eight Rooms Painted White or Yellow:
   Each room has 2 color choices.
   Total outcomes = $2^8 = 256$.

10. Six Books Arranged on a Shelf:
    Total outcomes = $6! = 720$.

Would you like detailed explanations for any specific problem? Here are some follow-up questions to enhance understanding:

1. What is the significance of the counting principle in these scenarios?
2. How does set notation simplify representing sample spaces?
3. Why is factorial important in problems like arranging books?
4. What are other real-world examples where counting principles apply?
5. How do combinations differ from permutations in these problems?

Tip: Always verify whether order matters in counting problems—it impacts whether to use permutations or