Count ways to make an amount with coins

2.MD.C.8 · adapt · grade 2

Archetype: Two-Category Counts from a Total · step in a 3-type progression

Mia has the coins shown below. To buy one toy that costs $20$ ¢, find how many different ways she can pay exactly the price of the toy.

Dimes (10¢)	Nickels (5¢)	Pennies (1¢)
2	2	10

Show solution

Understand

Mia has 2 dimes, 2 nickels, and 10 pennies. Count how many different combinations of these coins pay exactly 20 cents.

Givens

Available coins: 2 dimes (10 cents each), 2 nickels (5 cents each), 10 pennies (1 cent each).
The toy costs exactly 20 cents.
She must pay the exact price.

Unknowns

The number of different ways to make exactly 20 cents from her coins.

Constraints

She cannot use more dimes, nickels, or pennies than she actually has.
Two ways are different if they use different counts of any coin.

Plan

#2 Make a Systematic List · also uses: #8 Analyze the Units

This is a 'how many ways' question over a small, bounded set of coins, so I make an organized list ordered by the number of dimes, tracking the cent value of each coin to keep the total at 20.

Execute

#8 Analyze the Units 2.MD.C.8

Each dime is 10 cents, each nickel 5 cents, each penny 1 cent. I need combinations that total 20 cents, using at most 2 dimes, 2 nickels, and 10 pennies.

10d + 5n + 1p = 20

Turning every coin into its cent value makes the total easy to add up and compare to 20.

#2 Make a Systematic List 2.MD.C.8

Two dimes already make 20 cents, so no nickels or pennies are needed. That is 1 way: 2 dimes.

10 + 10 = 20

Two dimes hit 20 exactly, leaving nothing more to add.

#2 Make a Systematic List 2.MD.C.8

One dime is 10 cents, so I still need 10 cents from nickels and pennies. Options: 2 nickels (10 cents); 1 nickel and 5 pennies; 0 nickels and 10 pennies. Each respects her coin limits, giving 3 ways.

10 + (2\times5),\ 10 + 5 + (5\times1),\ 10 + (10\times1)

After one dime, the leftover 10 cents can be split among nickels and pennies in a few neat ways.

#2 Make a Systematic List 2.MD.C.8

With no dimes I need the full 20 cents from at most 2 nickels and 10 pennies. The most nickels and pennies give is 2 nickels (10 cents) plus 10 pennies (10 cents) = 20 cents, exactly one way. Using fewer nickels would need 15 or 20 pennies, but she only has 10, so those fail.

(2\times5) + (10\times1) = 20

With only 10 pennies on hand, she must use both nickels, leaving exactly one workable combination.

#2 Make a Systematic List 2.MD.C.8

Add the cases: 1 way with 2 dimes, 3 ways with 1 dime, 1 way with 0 dimes.

1 + 3 + 1 = 5

The cases by number of dimes don't overlap, so the total is just their sum.

Answer: 5 ways

Review

Each listed combination sums to exactly 20 cents and never exceeds her supply of 2 dimes, 2 nickels, and 10 pennies, so 5 distinct ways is consistent.

Build a table with columns for dimes, nickels, and pennies, fill in every row whose values total 20 cents within the limits, and count the rows.

Standards · min grade 2

2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies — Counting coin combinations that make exactly 20 cents.

💡 Organize your guesses by how many dimes you use, and the rest of the coins fall into place by twenties!