There are 5 people who need to share 8 dollars. Each person can get no more than 4 dollars. We want to know how many ways we can give out the 8 dollars.
We can use a special way of counting called "generating functions" to solve this problem. We start by making a list of all the possible ways to give out the dollars, where the first number is the number of dollars the first person gets, the second number is the number of dollars the second person gets, and so on. For example, one possible way is:
1, 1, 1, 2, 3
This means the first person gets 1 dollar, the second person gets 1 dollar, the third person gets 1 dollar, the fourth person gets 2 dollars, and the fifth person gets 3 dollars.
We can then make a special formula using these numbers, and we can use this formula to find the answer to our problem. The formula looks like this:
(1 + x + x^2 + x^3 + x^4)^5
We then need to find the coefficient of x^8 in this formula. This tells us how many ways we can give out 8 dollars to 5 people.
Using some math, we get the answer of 290. This means there are 290 ways to give out the 8 dollars.
So, there are 290 ways to give out 8 dollars to 5 people, where each person can get no more than 4 dollars.