A Card Game
I have a bag of cards numbered from 1 to 100 while my friend has a bag of cards numbered from 1 to 200. The goal is for me to successively pick two cards without replacement from my bag, such that the sum is equal to the single card that my friend picks from his bag. For example, if my friend picks up the card 4, then I must pick up \( (1,3) \) or \( (3,1) \). I cannot pick up \( (2,2) \).
Over all card choices, how many ways can the goal be met?
Note: Order does matter.