Solution
Making Change
Show the problem again
In how many ways can you make 25 cents using any number of pennies, nickels, and dimes?
Worked solution
The answer is 12. Case on the number of dimes. Zero dimes: 0–5 nickels (6 ways). One dime: 15 cents remain, 0–3 nickels (4 ways). Two dimes: 5 cents remain, 0–1 nickels (2 ways). Pennies fill the rest uniquely each time: 6 + 4 + 2 = 12 ways. (Generating functions systematize this for harder versions like changing a dollar.)
Source: Standard coin-partition counting exercise from introductory combinatorics texts. Statement written for AxiomIQ.