Question

Suzanne 去银行取了 $800。柜员用 $20、$50 和 $100 纸币给她这笔钱，并且每种面额至少有一张。Suzanne 可能收到多少种不同的纸币组合？

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?

$45$

$21$

$36$

$28$

$32$

答案：B

知识点：丢番图方程奇偶性基本计数

难度评级：1500

解答：

设 $a, b, c \ge 1$ 分别为 $\$20, \$50, \$100$ 纸币张数。则 $20a + 50b + 100c = 800$ ，化为 $2a + 5b + 10c = 80$ 。因为 $2a$ 和 $10c$ 都是偶数， $5b$ 也必须为偶数，所以 $b = 2t$ 。此时 $a = 40 - 5t - 5c \ge 1$ 条件等价于 $t + c \le 7$ ，其中 $t, c \ge 1$ 。这样的正整数对数量为 $1 + 2 + \cdots + 6 = 21$ 。所以正确答案是 B。

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 第 11 题

答案：B

难度评级：1500

解答：

其他年份的第 11 题