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Math Help - Money Distribution

  1. #1
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    Money Distribution

    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
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  2. #2
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    Quote Originally Posted by JoAdams5000 View Post
    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?
    You need to post something to show effort on your part.
    A) Hint: How many ways can we give $70 to three people?
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  3. #3
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    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
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  4. #4
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    The number of ways to put N identical items into k distinct cells is \binom{N+k-1}{N}=\frac{(N+k-1)!}{N!(k-1)!}.
    In this problem we have 100 one dollar bills and 3 different people.
    Give Bill 10 of them and Frank 20 of them. That leaves 70 of them to deal with.
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