2018 AMC 10B Problem 3

Below is the professionally curated solution for Problem 3 of the 2018 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 10B solutions, or check the answer key.

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Concepts:basic counting systematic listing

Difficulty rating: 950

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?

$2$

$3$

$4$

$6$

$24$

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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