Calculate number of dollar bill denominations to go into a cash bag

I am having trouble figuring what function (calculation) to use.

First, an overview of why I need this. I run fun promotions in schools. When a child sells widgets in a fundraiser, I let them draw cash out of a cash bag as a reward. Here are the exact instructions to the students...

For every 3 widgets you sell in our fundraiser, I will let you draw 1 bill out of a cash bag. The cash bag will be filled with $1's, $5's, $10's and $20's. Example: If you sell 9 widgets, you get THREE pulls out of the cash bag (one bill per draw).

Given the TOTAL NUMER OF DRAWS to be done within a school, I want an Excel formula (or calculation) to tell me how many $1's, $5's, etc to put into the cash bag.

As a general rule I want more lower denominations (such as $1's) and fewer high denominations (such as $20's).

I want to spend an exact predetermined dollar amount TOTAL on this promotion.

For example:

GIVEN: I need 891 draws at a school.
GIVEN: I want to spend $2,219 total.

How many $20 bills should I put in the bag?
How many $10 bills should I put in the bag?
How many $5 bills should I put in the bag?
How many $1 bills should I put in the bag?

I know that there could be MANY answers to this question, depending on exactly how many of each denomination I wish- but I don't really care about the exact amount. I am content to let Excel determine that for me as long as there are MORE lower denominations (such as $1's) and FEWER high denominations (such as $20's).

My brain just isn't big enough to make Excel work for me on this. Is it even possible?