mustish1
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50 dollars bill divide among seven peples
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.
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.
ASKER CERTIFIED SOLUTION
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ASKER
means
50*50*50*50*50*50*50
but how we get this number C(56,6)?
50*50*50*50*50*50*50
but how we get this number C(56,6)?
There are 56 items to arrange: 50 identical dollars and 6 identical partitions.
ASKER
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
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
SOLUTION
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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.
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.
ASKER
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
Tom
instead of 20.