At a fundraiser, there are 100 people. Each person has a distinct net worth. Everyone is paired with another person such that there are 50 pairs. The person with the lower net worth in the pair decides the amount that is to be donated and the person with the higher net worth has to donate double the amount eg: the person with lower net worth donates \$10 then the person with the higher net worth has to donate $20.
Assuming that all donations are whole dollars and that a person can donate a maximum of 9 million dollars. How many ways are there for all the different participants to donate money such that the total amount collected is $9 million?
My approach: I was unable to solve this question. But the way I think it should go is that we consider each pair to contribute $3x$ ($x$ from the lower net worth person and $2x$ from the higher net worth person). That means that we now have $3x_1, 3x_2, 3x_3,..., 3x_{50}$ such that $3x_1 + 3x_2 + 3x_3 + ... + 3x_{50} = 9$ million. We now imagine there are 50 boxes, the money in each box represents the money donated by the pair with the corresponding number. For every dollar that is donated, it has 50 different boxes it could have come from. This means that there $(90million)^{50}$ ways of donating the money. But I sure this answer over counts a lot.
While I understand that there are questions that seem similar to this one on this website, please understand that the ones I went through don't provide the intuition of tackling questions of a similar nature. They merely provide answers based on the context of the question. I would appreciate it if someone could provide me with the intuition regarding questions like this.