Combinatorics: Finding a Generating Function

[Shoney45]

Homework Statement

A national singing contest has 5 distinct entrants from each state. Use a generating function for modeling the number of ways to pick 20 semifinalists if:

a) There is at most 1 person from each state

b) There are at most 3 people from each state.

Homework Equations

N/A

The Attempt at a Solution

The first thing I am doing is to try and create a generating function for the problem as stated. Once I can do that,I can move onto the things that are asked in 'a' and 'b'. But creating generating functions is still unclear to me.

So I think the five entrants from each state represent x (sub 1) through x (sub 5). Since there are fifty states, everything is raised to the fiftieth power, and the whole things equal 20 such that (1 + x + x^2 + x^3 + x^4 + x^5)^50 = 20.

I guess I'll stop there until I find out if I am correct or not.

 

[mighty2000]

Thank you for your post. I can provide some guidance on creating a generating function for this problem.

First, let's define some variables:
- Let x represent the number of people from each state that can be chosen for the semifinals.
- Let n represent the number of states (in this case, n=50).

a) In order to have at most 1 person from each state, we can use the following generating function:
(1+x)^n = (1+x)^50
This represents the number of ways to choose x people from each state, with a maximum of 1 person from each state.

b) In order to have at most 3 people from each state, we can use the following generating function:
(1+x+x^2+x^3)^n = (1+x+x^2+x^3)^50
This represents the number of ways to choose x people from each state, with a maximum of 3 people from each state.

I hope this helps you in creating the generating function for this problem. Let me know if you have any further questions.