Let me break down the two problems from your image and provide solutions.

1. Five Points on a Circle: How Many Triangles Can Be Drawn?

To determine how many triangles can be formed by connecting any three of the five points on a circle, we calculate the combination of 5 points taken 3 at a time.

The formula for combinations is: $C(n, r) = \frac{n!}{r!(n-r)!}$ Here, $n = 5$ and $r = 3$: $C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \cdot 4 \cdot 3!}{3! \cdot 2 \cdot 1} = 10$

Thus, 10 triangles can be formed by connecting three points out of five on the circle.

2. Using Digits $0, 1, 2, 3, 4, 5$: How Many 3-Digit Numbers ≥ 200 Can Be Formed?

To solve this, consider:

• The digits available: $0, 1, 2, 3, 4, 5$.
• The condition: The 3-digit number must be ≥ 200.

Step 1: First digit (hundreds place)

The first digit must ensure the number is at least 200. Therefore, the possible values for the hundreds digit are $2, 3, 4, 5$ (4 choices).

Step 2: Second digit (tens place)

The second digit can be any of the 6 available digits ($0, 1, 2, 3, 4, 5$), since repetition is allowed (6 choices).

Step 3: Third digit (ones place)

Similarly, the third digit can also be any of the 6 available digits (6 choices).

Total combinations

Multiply the choices for each digit: $4 \cdot 6 \cdot 6 = 144$

Thus, 144 three-digit numbers ≥ 200 can be formed using the given digits with repetition allowed.

Questions for Exploration:

1. How would the result change if repetition of digits were not allowed in the second problem?
2. What if the condition in the second problem changed to numbers being "strictly greater than 300"?
3. How can the approach to the triangle problem be extended to calculate polygons with $k$ sides?
4. How would the triangle problem change if the points were not on a circle?
5. What are some real-world applications of these types of counting problems?

Tip:

When solving counting problems, always start by clearly identifying restrictions and constraints, as this simplifies the calculations significantly!