There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10  rows of the triangle?

To solve the problem of finding how many even integers are in the top 10 rows of Pascal's Triangle, we can analyze the distribution of even and odd numbers in Pascal's Triangle.

### Step-by-Step Solution:

1. **Pascal's Triangle Overview**: Pascal's Triangle is a triangular array of the binomial coefficients. The \(n\)th row (starting from \(n = 0\)) consists of the coefficients of the binomial expansion \((x + y)^n\).

2. **Even and Odd Integers**: An entry in Pascal's Triangle is even if and only if the corresponding binomial coefficient is even.

3. **Observing the Pattern**:
   - **Row 0**: \( [1] \) (1 odd)
   - **Row 1**: \( [1, 1] \) (2 odd)
   - **Row 2**: \( [1, 2, 1] \) (1 even)
   - **Row 3**: \( [1, 3, 3, 1] \) (0 even)
   - **Row 4**: \( [1, 4, 6, 4, 1] \) (2 even)
   - **Row 5**: \( [1, 5, 10, 10, 5, 1] \) (2 even)
   
   From this pattern, the rows begin to exhibit a mix of even and odd numbers. The symmetry in Pascal's Triangle and Lucas' Theorem implies that the number of even entries increases in a non-trivial way.

4. **Counting Even Numbers in the Top 10 Rows**:
   We know there are 4 even integers in the top 5 rows. The pattern indicates an increase in the number of even integers as the rows increase.

   - **Row 6**: 4 even integers
   - **Row 7**: 4 even integers
   - **Row 8**: 4 even integers
   - **Row 9**: 4 even integers
   
   So, in the top 10 rows (rows 0 to 9), we can count:
   - Even integers in rows 0 to 4: 4 even integers
   - Even integers in rows 5 to 9: 4 per row × 5 rows = 20 even integers
   
   **Total Even Integers in Top 10 Rows**: \(4 + 20 = 24\) even integers.

Thus, there are **24 even integers** in the top 10 rows of Pascal's Triangle.

### Follow-Up Questions:
1. How many odd integers are in the top 10 rows of Pascal's Triangle?
2. Can you derive a general formula for counting even numbers in any row of Pascal's Triangle?
3. How does Lucas' Theorem relate to the distribution of even and odd numbers in Pascal's Triangle?
4. What is the significance of the Sierpiński triangle pattern in Pascal's Triangle when considering even and odd numbers?
5. How would the count of even numbers change in the top 20 rows of Pascal's Triangle?

**Tip**: Pascal's Triangle is not only useful for combinatorial coefficients but also for exploring number theory concepts like divisibility and modular arithmetic.

Explore how to count even integers in the top 10 rows of Pascal's Triangle using combinatorial coefficients and Lucas' Theorem. Detailed step-by-step solution provided. Suitable for advanced high school students.

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