If X is the sum of the first 50 positive even integers and Y is the sum of the first 50 positive odd integers...

This seems to arithmetic progression problem but it can be solved in other simple way.

Information available
x = sum of 50 even numbers => (2 + 4 + 6 .... + 100)
y = sum of 50 odd numbers => (1 + 3 + 5 ... + 99)

To deduce x - y, now let's break series
x - y (break up)
=> 2 - 1 => 1
=> 4 - 3. => 1
=> 6 - 5. => 1
....
...
=> 100 - 99 => 1

So after each subtraction, we get 1 as the value. So there total 50 number in series, hence 50 * 1 => 50.
Choose C