Question

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 using any arrangement of gold and silver coins, assuming order matters?

$3$

$7$

$10$

$13$

$16$

Video solution:

Click to load, then click again to play

Written solution:

Count by the number of silver coins. With no silver coins, there is $1$ stack. With one silver coin, there are $5$ gold coins and $1$ silver coin, so $6$ arrangements. With two silver coins, there are $2$ gold coins and $2$ silver coins, so $\binom42=6$ arrangements. More silver coins are too thick. The total is $1+6+6=13$ .

2026 AMC 8 Problem 20

Video solution:

Written solution:

Problem 20 in Other Years