2018 AMC 10B Problem 3

Problem 3 of 25EasierCombinatorics Problem-Solving Techniques

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,

4,

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

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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.

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Tagged: basic counting · systematic listing

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