Let's first focus on the two problems shown in the image. I'll extract and summarize them before solving:

Problem 43:

• Text: "In a show, there are 8 people in a dance. How many groups of 4 people can be chosen to perform? Use the expression $\frac{8!}{4!(8-4)!}$ to find how many groups can be chosen."
• Type: This is a combinatorics problem, asking for how many ways we can choose 4 people out of 8.

Problem 45:

• Text: "The front door of a house is painted white and blue. Each window is a square with a side length of 4 inches. What is the area of the door that is painted blue?"
• Type: This is a geometry problem, asking for the area of the blue-painted sections of a door with windows, where each window has a given side length.

Let's solve these step-by-step.

Problem 43 Solution:

The problem gives us the formula for combinations: $\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}$ We can calculate this as follows.

Problem 45 Solution:

The windows are squares with a side length of 4 inches. The area of a square is given by: $\text{Area of one square} = 4 \times 4 = 16 \text{ square inches}$ Since there are three windows painted blue (as seen in the image), we multiply by 3 to find the total blue-painted area: $\text{Total blue area} = 3 \times 16 = 48 \text{ square inches}$

I'll now calculate the value for Problem 43.### Problem 43 Answer: The number of ways to choose 4 people out of 8 is: $\binom{8}{4} = 70$

Thus, there are 70 different groups of 4 people that can be chosen.

Problem 45 Answer:

The total area of the blue-painted sections of the door is: $48 \text{ square inches}$

Let me know if you would like more details on any of these solutions!

Here are 5 related questions to expand your understanding:

1. How would the answer to Problem 43 change if we had 10 people and needed to choose groups of 5?
2. What would the total area be if each window in Problem 45 had a side length of 5 inches?
3. How would the result for combinations change if repetition of people were allowed in Problem 43?
4. How could you calculate the number of possible groups if people could be chosen in different sizes (e.g., groups of 3, 4, or 5)?
5. If the door had 5 windows, how would that affect the blue-painted area?

Tip: When dealing with combinations and permutations, always double-check whether repetition or order matters in the problem!