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?
We can list the first 10 rows of Pascal's triangle, and mark the even numbers.

[asy]
usepackage("amsmath");
unitsize(0.5 cm);

int i, j, n;

for (int i = 0; i <= 9; ++i) {
for (int j = 0; j <= 9; ++j) {
  if (i + j <= 9) {
    n = choose(i + j,i);
    if (n % 2 == 0) {label("$\boxed{" + string(n) + "}$", i*(-1,-1) + j*(1,-1));}
    if (n % 2 == 1) {label("$" + string(n) + "$", i*(-1,-1) + j*(1,-1));}
  }
}}
[/asy]

Thus, the number of even numbers is $1 + 3 + 2 + 3 + 7 + 6 = \boxed{22}.$