Correct Answer - `(10)/(19)`

The total number of ways in which 2 integers can be chosen from the given 20 integers is `.^(20)C_(2)`.

The sum of the selected numbers is odd if exactly one of them is even and only one is odd. Therefore, the favorable number of outcomes is `(.^(10)C_(1)xx.^(10)C_(1))/(.^(20)C_(2)) = (10)/(19)`