2026 AMC 8 Problem 20: Solution

Work through 2026 AMC 8 Problem 20 on combinatorics with a written solution, hints, and a visual explanation.

Counting & ProbabilityLevel 4 · Challenge

The land of Catania uses gold coins and silver coins. Gold coins are 1 mm thick and silver coins are 3 mm thick. In how many ways can Taylor make a stack of coins that is 8 mm tall, assuming order matters?

  • A. 3
  • B. 7
  • C. 10
  • D. 13
  • E. 16
Hints
  1. Set up an equation relating the number of gold and silver coins to the total stack height.
  2. Find all whole-number combinations of gold and silver coins that add up to exactly 8 mm.
  3. For each combination, count the distinct orderings (since order matters), then add the counts from all combinations together.
Read the step-by-step solution

Answer: D · 13

  1. Find the coin combinations

    Let G gold coins (1 mm) and S silver coins (3 mm). The heights must add to 8: G + 3S = 8. The whole-number solutions are (G,S) = (8,0), (5,1), (2,2).

    G + 3S = 8

  2. Count the orderings in each case

    Order matters, so count the distinct stacks in each case. (8,0): only 1 way. (5,1): C(6,1) = 6 ways. (2,2): C(4,2) = 6 ways.

  3. Add up all the cases

    The three cases are disjoint, so add: 1 + 6 + 6 = 13 possible stacks.

    1 + 6 + 6 = 13

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