Competition · AMC preparation · step 4 of 4

AMC 8 · 2020 · #10

Grade 7 counting

Archetype Overlap / Complement Counting └ Sub-archetype Complement by Subtracting Forbidden Arrangements

permutations-basiccomplementary-counting complementary-counting ↑ Prerequisites: systematic-enumeration

📏 Medium solution 💡 3 insights

Problem

Zara has 4 distinct marbles — Aggie (A), Bumblebee (B), Steelie (S), Tiger (T) — and lines them up on a shelf. She wants to count the line-ups in which Steelie and Tiger are NOT next to each other. How many such arrangements are there?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

The word 'not next to' is the classic trigger for Tool #16 (Complement). Counting arrangements where S and T avoid each other directly forces messy case-splitting by where S goes; counting the OPPOSITE event (S and T together) is much easier because we can glue them into a single block. We'll use Tool #2 (Systematic List) as a sanity check for the small total of 24 arrangements, and Tool #3 (Eliminate) to confirm the answer matches one of the given choices.

1STEP 1

Count all arrangements

With no restriction, filling the 4 positions in order gives 4 × 3 × 2 × 1 = 24 total line-ups by the multiplication rule.

Total = 4 × 3 × 2 × 1 = 24

2STEP 2

Count the block arrangements

Glue Steelie and Tiger into one block [ST]; arranging that block with Aggie and Bumblebee gives 3 × 2 × 1 = 6 orders.

Block orders = 3! = 3 × 2 × 1 = 6

3STEP 3

Double for the two inside orders

Inside the block, ST or TS gives 2 internal orders, so forbidden line-ups total 6 × 2 = 12.

Forbidden = 6 × 2 = 12

4STEP 4

Subtract the forbidden arrangements

Subtract the forbidden line-ups from the total to leave the ones with Steelie not beside Tiger: 24 - 12 = 12.

Allowed = 24 - 12 = 12

The complement rule: (what we want) = (everything) - (what we don't want).

The line-ups where Steelie and Tiger are not next to each other number 24 minus 12, which is 12.

▸ Why?

Every line-up of the four marbles falls into exactly one of two cases — Steelie and Tiger adjacent, or not adjacent — with no line-up left out and none counted in both, so the not-adjacent ones are the whole collection with the adjacent ones taken out.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The whole collection holds 24 line-ups: the four spots are filled one at a time with 4, then 3, then 2, then 1 choices, and those independent choices multiply to 24.

Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.

▸ Why?

The line-ups with Steelie and Tiger adjacent number 12.

▸ Why?

Gluing Steelie and Tiger into one block, each adjacent line-up matches exactly one arrangement of the three units — the block, Aggie, Bumblebee — together with one of the block's two inside orders, so counting those counts the adjacent line-ups exactly.

▸ Why?

Pairing every adjacent line-up with one block-arrangement-and-inside-order, and each of those back to one adjacent line-up, leaves nothing unmatched, so the two collections are the same size.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

The three units fill three spots with 3 choices, then 2, then 1, and those choices multiply to 6 orders.

Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.

▸ Why?

The 6 orders of the units and the block's 2 inside orders are independent choices, so multiplying them gives 6 times 2, which is 12.

Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.

4.OA.A.3Change Focus Count The Complement

5STEP 5

Match against the choices

Among the choices only (C) equals 12, so the answer is (C).

12 = (C)

Answer

The total 24 is exactly 4!, which is correct for 4 distinct objects in a row. The forbidden count 12 is half of 24, which makes intuitive sense: among 4 random items, the probability that two specific items are adjacent is

\frac{2 · 3!}{4!} = \frac{12}{24} = \frac{1}{2}

, so 'not adjacent' is also

\frac{1}{2}

, giving 12 good arrangements. The answer 12 is in the middle of the choices (6, 8, 12, 18, 24), which is the expected magnitude for a balanced complement count.

💡Key takeaway

This AMC 8 problem only needs Grade 7 counting with organized lists you already know — count everything, count the bad cases, then subtract!

Count all arrangements
Count the block arrangements
Double for the two inside orders
Subtract the forbidden arrangements
Match against the choices

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Count all arrangements

Count the block arrangements

Double for the two inside orders

Subtract the forbidden arrangements

Match against the choices

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