To solve this problem, we need to identify the number of even integers in the top rows of Pascal's Triangle. Pascal's Triangle is a triangular array of binomial coefficients, where each entry ${n \choose k}$ represents the binomial coefficient for a given $n$ and $k$.

Step 1: Understanding the Problem

We know that there are four even integers in the top five rows of Pascal's Triangle. Pascal's Triangle is built with rows indexed from $n = 0$ upwards:

[ \begin{align*} \text{Row 0:} & ; 1 \ \text{Row