Question

Suzanne went to the bank and withdrew $800. The teller gave her this amount using $20 bills, $50 bills, and $100 bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?

$45$

$21$

$36$

$28$

$32$

Answer: B

Concepts:Diophantine Equation parity basic counting

Difficulty rating: 1500

Solution:

Let $a, b, c \ge 1$ count the $\$20, \$50, \$100$ bills. Then $20a + 50b + 100c = 800,$ which divides down to $2a + 5b + 10c = 80.$ Both $2a$ and $10c$ are even, so $5b$ is too, forcing $b = 2t.$ Now $a = 40 - 5t - 5c \ge 1$ means $t + c \le 7.$ With $t, c \ge 1,$ the pairs number $1 + 2 + \cdots + 6 = 21.$ Thus, B is the correct answer.

2023 AMC 10B Problem 11

Answer: B

Difficulty rating: 1500

Solution:

Problem 11 in Other Years