Problem
Count ways to make an amount with coins
Mia has 2 dimes, 2 nickels, and 10 pennies. In how many different ways can she pay exactly the 20-cent price of a toy?
Measurement & data
Your answer
ways
Watch out You cannot use more coins than she owns. There are only 10 pennies to spend.
How to solve
1STEP 1
Set up the goal as an amount
Turn coins into amounts and find combos that total 20 cents — without exceeding what she has (2, 2, 10).
10 × a + 5 × b + 1 × c = 20
Each coin has a fixed value, so turning counts into amounts makes the total easy to add and compare to the target.
Make Systematic List2.MD.C.82STEP 2
Case: 2 dimes
Two dimes already make 20 cents — nothing else needed, so that's 1 way.
10 + 10 = 20
Filling up on the biggest coin hits the target exactly, leaving nothing to add, so this case pins one way.
Make Systematic List2.MD.C.83STEP 3
Case: 1 dime
One dime leaves 10 cents to fill with nickels and pennies — 3 ways to do it.
10 + 5 × 2, 10 + 5 + 1 × 5, 10 + 1 × 10
Fixing the dime fixes the leftover exactly, so we only count the few ways smaller coins split that leftover. Those three ways are: 2 nickels; 1 nickel plus 5 pennies; or 10 pennies.
Make Systematic List2.MD.C.84STEP 4
Case: 0 dimes
With no dimes, only 10 pennies exist, so both nickels must be used to reach 20 cents — 1 way.
5 × 2 + 1 × 10 = 20
The 10-penny supply is capped, so using fewer nickels would need more pennies than she has, leaving exactly one workable combo.
Make Systematic List2.MD.C.85STEP 5
Add up all the ways
Add up the 2-, 1-, and 0-dime cases for 5 ways.
1 + 3 + 1 = 5
Sorting every way by dime count (0, 1, 2) makes three groups that miss nothing and share nothing, so the group counts add up to the total.
Make Systematic List2.MD.C.8The total number of ways is the sum of the ways counted in each separate case grouped by how many dimes are used.
Why?
Sorting every payment by how many dimes it uses breaks all the ways into three groups that miss nothing and share nothing, so the group counts simply add up to the total.
Why?
A payment can use only 0, 1, or 2 dimes since Mia has just 2 dimes, and each payment has one fixed number of dimes, so every way falls into exactly one of the three groups with none left out and none counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
5 ways
1 + 3 + 1 = 5
Check — all 5 combos total exactly 20 cents and never exceed 2 dimes, 2 nickels, and 10 pennies.
Takeaway
Order your list by how many of the biggest coin you use, and the leftover amount falls into place.
- 2 dimes → 1 way
- 1 dime → 3 ways
- 0 dimes → 1 way, add up to 5 ways
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