How many positive integer solutions are there for the equation x + y = 20, where x and y are integers greater than or equal to 1?
Since both x and y must be positive integers (greater than or equal to 1), we can rewrite the equation as:

x = 20 - y

Now, let's find all possible values of x and y that satisfy this equation. We know that x and y must be between 1 and 19 (inclusive) because if either x or y were 20 or larger, the other variable would have to be zero or negative, which is not allowed.

There are 19 possible values for x (from 1 to 19) and for each value of x, there is a corresponding value of y such that x + y = 20. Therefore, there are 19 positive integer solutions for the equation x + y = 20.