Number of objects distributed between four people

[gruba]

Homework Statement

Four people are dealing the total amount of money, which is 1000 monetary units in terms of 100 monetary units. Count the number of ways for this distribution if:
1) Every person doesn't have to get any money
2) Every person will get at least 100 monetary units
3) First person will get at least 500 m.u. and other three people at least 100 m.u.

Homework Equations

-Combinatorics

The Attempt at a Solution

The problem doesn't state what is the maximum amount of money that each person can get.
Assuming that, in 1) every person will get the same amount (200 m.u.), the total number of counts would be the coefficient with x^{1000} in (1+x+...+x^{200})^4.

In 2), assuming that the maximum amount for every person is 200 m.u, the total number of counts would be the coefficient with x^{1000} in (x^{100}+...+x^{200})^4.

In 3)\Rightarrow x^{500}(x^{100}+...+x^{200})^3

What do you think, how to solve this problem?

 

[andrewkirk]

You can convert (2) and (3) into the same problem as (1) with a different total to be distributed, by doing something non-probabilistic before you start the random part. What might that non-probabilistic step be?BTW, your formula $(1+x+...+x^{200})^4$ doesn't need to be have such a huge number of terms. Since everything is done in multiples of 100mu, divide the amounts by 100mu before you start.

 

[Ray Vickson]

Homework Statement

Four people are dealing the total amount of money, which is 1000 monetary units in terms of 100 monetary units. Count the number of ways for this distribution if:
1) Every person doesn't have to get any money
2) Every person will get at least 100 monetary units
3) First person will get at least 500 m.u. and other three people at least 100 m.u.

Homework Equations

-Combinatorics

The Attempt at a Solution

The problem doesn't state what is the maximum amount of money that each person can get.
Assuming that, in 1) every person will get the same amount (200 m.u.), the total number of counts would be the coefficient with x^{1000} in (1+x+...+x^{200})^4.

In 2), assuming that the maximum amount for every person is 200 m.u, the total number of counts would be the coefficient with x^{1000} in (x^{100}+...+x^{200})^4.

In 3)\Rightarrow x^{500}(x^{100}+...+x^{200})^3

What do you think, how to solve this problem?

I found the wording of your problem confusing: if you mean that each person can receive 0 or 100 or 200 or ... (i,e, multiples of 100) then you should say that a bit more clearly. If that is what you mean, it would be much easier to let 100 m.u be a new monetary unit. Now the amount each person can receive is an integer number of these new units and we have a total of 10 new units for distribution.

Also, to clarify: in (3), do you mean (in these new units) that persons 2--4 get at least 1 in total, or that persons 2--4 will get at least 1 each?

 

[haruspex]

1
Assuming that, in 1) every person will get the same amount (200 m.u.)

That would not be a good assumption. As I read it, the whole amount is to be divvied up, and there's no restriction on who gets what. Essentially, you have ten identical things to put in four (distinct) baskets.