Find the count of possible bank-note splits for 4 people, when you want to split \$150 for each person with bank-notes of values \$10, \$20 and \$50.
- What is the count of splits when each person will the amount split differently?
- What is the count of splits when there are no limitations?
Could someone please explain me how to solve this? I think it should be done with using basic combinatorics rules and Diophantine equation which I am a little familiar with.
Example: One of many solutions for (1) (each person has the amount in different bank-note spitls)
- Person A - $3\cdot\$50 + 0\cdot\$20 + 0\cdot\$10$
- Person B - $2\cdot\$50 + 2\cdot\$20 + 1\cdot\$10$
- Person C - $2\cdot\$50 + 1\cdot\$20 + 3\cdot\$10$
- Person D - $2\cdot\$50 + 0\cdot\$20 + 5\cdot\$10$
Example: One of many solutions for (2) (the splits can be the same for earch person)
- Person A - $3\cdot\$50 + 0\cdot\$20 + 0\cdot\$10$
- Person B - $3\cdot\$50 + 0\cdot\$20 + 0\cdot\$10$
- Person C - $3\cdot\$50 + 0\cdot\$20 + 0\cdot\$10$
- Person D - $3\cdot\$50 + 0\cdot\$20 + 0\cdot\$10$
Disclaimer: this is not a homework
Using generating functions we will look at the number of ways to split \$150 for one person. Since we are allowed to use only the following denominations \$10, \$20 and \$50, the generating function for the problem (read the link I posted, it's important to understand how this technique works) becomes: $$\frac{1}{(1-x^{10})(1-x^{20})(1-x^{50})}$$ The coefficient of $x^{150}$ is the answer. I am going to cheat, the answer is $18$.
There are 4 persons involved, the answer is $18^4$.