2023 AMC 10B Problem 11

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

Divide

20a + 50b + 100c = 800

by

10;

parity then forces the number of

\$50

bills to be even

Writing

b = 2t,

count positive integer pairs

(t, c)

with

t + c \le 7

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.

Problem 11 in other years