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50 dollars bill divide among seven peples

mustish1
mustish1 asked
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Snow White has 50 one-dollar bills, which she wishes to divide up among seven different dwarves. Each dward may receive any (integra) number of bills, from 0 to 50. How many different ways can she distribute this money?

My logic is 51*51*51*51*51 cauz its from 0 to 50

but the answer is C(56,6)=32,468,436
I think here 6 means 0 + 5 times 51. I may wrong needs help

Thanks.
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This is just like the last question, but now the seven numbers add up to 50
instead of 20.
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Imagine you have 50 dollar bills (or marbles) lined up in a row.
And you have six partitions (dividers) to place in the row as well.
Six partitions make seven boxes, or dwarves, or variables.

Author

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means
50*50*50*50*50*50*50

but how we get this number C(56,6)?
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There are 56 items to arrange: 50 identical dollars and 6 identical partitions.

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There are 7 dwarfs and the number of their dollar bills add up to 50.

y1+y2+...+y7 = 50

If i replace each y by x-1, i have the equation
x1-1+x2-1+...+x7-1=50

I think im doing wrong or in a hard way
The analogy of lining up the bills sort of works.  Take 56 bills and line them up.  Take 6 markers and put them on top of any 6 different bills.  Everything to the left of the first marked bill, give to the first dwarf.  Everything between the first and second marked bill, give to the second dwarf.  Everything to the right of the last marked bill, give to the seventh dwarf.  Keep the 6 marked bills for yourself.  The reason for the extra six marked bills is so that you can have two marked bills in a row thus allowing for giving $0 to any dwarves.  That wouldn't work with the partition scheme.  So that's why C(56,6) instead of C(51,6) which the partitions would suggest.

The formula for picking k things out of n things is n! / k!(n-k!)

So,

C(56,6) = 56! / (56-6)!6! or 51*52*53*54*55*56/720.  

But I get 46,754,547,840.
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>>  The analogy of lining up the bills sort of works.

It is much more than that.
It is the classic way to teach partition theory applications in statistical mechanics.
    How many ways are there to partition quantized energy among N particles.

You definitely don't need or want to add extra marked bills.  The bills and the partitions are separate entities.

You can place all of the partitions to the left of the first bill, giving $0 to the first six dwarves and $50 to the seventh.

You can place three partitions between the first and second bill, and three before the last.
 Giving $1 to dwarves 1 and 7, $48 to dwarf 4, and $0 to the rest.

Every combination of bills and partitions represents a possible distribution/solution.

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Thanks d-glitch
Note that it can also be generalized. You could start with $1 and 1 dwarf. Then try $1 and 2 dwarves, $2 and 2 dwarves and $2 and 3 dwarves. By using small numbers, it might be easier to see how the same generalized formula continues to work as the numbers get larger.

Tom