How many ways are there to distribute $100 among three people, Bill, Frank and Joe, such that:
a) Bill gets at least $10 and Frank gets at least $20?
b) Same as above but now we require that Joe gets at most $4?
I am having some trouble getting started on this problem. I would appreciate any suggestions on how to solve this. Thanks
My attempt at a solution for A):
suppose A starts off with $10, and B starts off with $20 (so they will already satisfy the 2 conditions) and find how many ways you can distribute the rest of the $70.
A B C
0 0 70
0 1 69
0 2 68
..
..
0 70 0
so when A is fixed with $0, there are 70 ways of distributing the $70 between B and C
A B C
1 0 69
1 1 68
..
..
1 69 0
so when A is fixed with $1, there are 69 ways to distribute the rest of the $69
..
..
continuing the pattern for A fixed with $n, there is 70-n ways to distribute the $(70-n) between B and C
thus, the sum of all these possibilities is 70 + 69 + .. + 2 + 1 = 70(1+70) / 2 = 2485 ways
Does this solution make any sense for A)?
I did something similiar for B)...but am having trouble including the new condition of $4.
Thanks for your help