2018 AMC 10B Problem 3

Question

In the expression

(\underline{\phantom{x}} \times \underline{\phantom{x}}) + (\underline{\phantom{x}} \times \underline{\phantom{x}})

each blank is to be filled in with one of the digits

1,

2,

3,

or

4,

with each digit being used once. How many different values can be obtained?

2

3

4

6

24

Answer: B

Concepts:basic counting systematic listing

Difficulty rating: 950

Small hint:

Neither multiplication nor the final addition depends on order, so only the way the four digits are split into two pairs matters

Big hint:

List the three ways to split

\{1,2,3,4\}

into two pairs

Solution:

Order inside a product doesn’t matter, and neither does the order we add the two products. So all that matters is how the four digits split into two pairs. There are three splits:

1\cdot2 + 3\cdot4 = 14,

1\cdot3 + 2\cdot4 = 11,

and

1\cdot4 + 2\cdot3 = 10.

That’s

3

different values. Thus, B is the correct answer.