Let's say that I have $n$ people. I have $m$ dollars, and I want to create bills of any number of dollars, such that the total value of the bills sum to $m$. I want to ensure that this set of bills is sufficient so that all distributions of money among the $n$ people can be satisfied by the set. What is the minimum number of bills that must be printed?
For example, if I have 5 people and 8 dollars, I can have the following distribution: 4 \$1 bills and 2 \$2 bills, which would amount to an answer of 6. This would be able to satisfy all distributions of coins, i.e. (1,1,1,1,4), (1,1,1,2,3), (1,1,2,2,2), (0,1,1,1,5), etc.
By satisfy I mean that the bills can be given to the people with each person having the right amount of money. Of course, you can't split bills.
If $m\leq n$, then I think the answer would just be $m$. Otherwise, I'm not so sure.