AMC 10 · 2018 · #3

Grade 3 algebra
Archetype Branching State Enumeration └ Sub-archetype Numeric Operation Tree
set-partitionmental-arithmetic systematic-enumeration ↑ Prerequisites: mental-arithmetic
📏 Medium solution 💡 2 insights
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Problem
The four blanks in ( × )+( × ) get the digits 1,2,3,4, each used exactly once. Count how many different totals the expression can equal.

Pick an answer.

(A)
2
(B)
3
(C)
4
(D)
6
(E)
24

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

A "how many different values" question with a small finite setup points straight at Tool #2 (Make a Systematic List). But listing all 4! = 24 ways to fill the blanks would be wasteful, so first apply Tool #15 (Organize Information in More Ways): because both × and + are commutative, the only thing that matters is how the four digits split into two pairs. That reframing turns 24 messy arrangements into just a handful of pairings to list. Tool #3 (Eliminate Possibilities) then confirms the count against the answer choices.

1STEP 1

Order inside and between products does not matter

Since a× b = b× a and P+Q = Q+P, the total depends only on how the four digits split into two pairs, not on any order.

(a× b)+(c× d) = (b× a)+(d× c) = (c× d)+(a× b)
2STEP 2

List the ways to split into two pairs

Fix digit 1 and choose its partner from 2, 3, or 4; the rest pair up automatically, giving three splits with none repeated or missed.

{1,2} & {3,4} {1,3} & {2,4} {1,4} & {2,3}
3STEP 3

Multiply each pair

For each split, multiply the two digits in each pair to get the two products that will be added.

1× 2=2, 3× 4=12; 1× 3=3, 2× 4=8; 1× 4=4, 2× 3=6
4STEP 4

Add the two products in each split

Add each split's two products to get its total: 14, 11, and 10.

2+12=14; 3+8=11; 4+6=10
5STEP 5

Count the distinct totals

The totals 14, 11, 10 are all different, so there are three distinct values — choice (B), not the order-counting traps 6 or 24.

{14, 11, 10} → 3 distinct values → (B)
Answer
3
The big choices 24 and 6 are traps: 24 = 4! counts every ordered filling, and 6 over-counts pairings; both ignore that × and + are commutative. There are genuinely only 3 ways to split four digits into two pairs, and the three totals 14, 11, 10 happen to be distinct, so no further merging occurs. Three distinct values, choice (B), is consistent.
💡Key takeaway

Because × and + don't care about order, the only choice is how to split 1,2,3,4 into two pairs — and there are just three ways, giving 14, 11, 10.

  • Order inside and between products does not matter
  • List the ways to split into two pairs
  • Multiply each pair
  • Add the two products in each split
  • Count the distinct totals

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