AMC 8 · 2003 · #16

Grade 4 counting

Archetype Casework by Position / Vertex Type └ Sub-archetype Vertex or Element Role

permutations-basiccombinations-basicsystematic-enumeration caseworksystematic-enumeration ↑ Prerequisites: multi-digit-arithmeticsystematic-enumeration

📏 Short solution 💡 2 insights

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Problem

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seats. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Ali, Bonnie, Carlo, and Dianna split into the four seats of a car: $1$ driver seat, $1$ front passenger seat, and $2$ back seats. Only Bonnie and Carlo can drive. Count every possible seating arrangement.

Givens: Four people: Ali, Bonnie, Carlo, Dianna; Four labeled seats: driver, front passenger, back-left, back-right; Only Bonnie and Carlo are allowed in the driver seat; The other three seats have no restriction; Answer choices: (A) $2$, (B) $4$, (C) $6$, (D) $12$, (E) $24$

Unknowns: The total number of valid seating arrangements

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #13 Count Smartly

One seat has a special rule (driver) and the other three are unrestricted. Tool #7 (Identify Subproblems) says: handle the constrained seat first, then handle the rest as a separate, simpler subproblem. With the driver picked, the remaining seats become "arrange $3$ people in $3$ labeled seats," which Tool #13 (Count Smartly) finishes with the multiplication principle: multiply the number of choices at each step. Filling the most-restricted slot first is the standard counting move because it prevents over-counting later.

Execute — Answer: D

#7 Identify Subproblems 4.OA.A.3 Step 1

Subproblem 1 — pick the driver.
The rule restricts the driver seat to Bonnie or Carlo.
That gives exactly $2$ choices for who drives.

$$\#\text{drivers} = 2 \;\; (\text{Bonnie or Carlo})$$

💡 Always start with the slot that has the fewest options — it locks in the hardest piece first.

#13 Count Smartly 4.OA.A.3 Step 2

Subproblem 2 — pick the front passenger.
After one person sits in the driver seat, $3$ people remain.
Any of them can take the front passenger seat, so there are $3$ choices.

$$\#\text{front passengers} = 3$$

💡 With no rule blocking anyone, the count is simply "how many people are left."

#13 Count Smartly 4.OA.A.3 Step 3

Subproblem 3 — pick the back-left passenger.
Two people remain after the driver and front passenger are seated.
Either of them can take the back-left seat, so $2$ choices.

$$\#\text{back-left} = 2$$

💡 Each filled seat shrinks the leftover pool by one — the choices shrink in lockstep.

#13 Count Smartly 4.OA.A.3 Step 4

Subproblem 4 — pick the back-right passenger.
Only $1$ person is left, so that person takes the last seat.
$1$ choice.

$$\#\text{back-right} = 1$$

💡 The last seat is forced once the other three are filled.

#13 Count Smartly 4.OA.A.3 Step 5

Combine the subproblems.
The four choices are independent (each pick is a separate decision), so multiply the option counts together.

$$2 \times 3 \times 2 \times 1 = 12 \;\Rightarrow\; \textbf{(D)}$$

💡 The multiplication principle: when a task splits into independent steps, total arrangements $=$ product of choices at each step.

[1] #7 4.OA.A.3 Subproblem 1 — pick the driver. The rule restricts the driver seat to Bonnie or

[2] #13 4.OA.A.3 Subproblem 2 — pick the front passenger. After one person sits in the driver sea

[3] #13 4.OA.A.3 Subproblem 3 — pick the back-left passenger. Two people remain after the driver

[4] #13 4.OA.A.3 Subproblem 4 — pick the back-right passenger. Only $1$ person is left, so that p

[5] #13 4.OA.A.3 Combine the subproblems. The four choices are independent (each pick is a separa

Review

Reasonableness: Without the driver restriction, $4$ people in $4$ labeled seats would give $4 \times 3 \times 2 \times 1 = 24$ arrangements — that is choice (E). The driver rule cuts the valid choices for one seat from $4$ down to $2$, exactly halving the total: $24 \div 2 = 12$. The answer $12$ matches choice (D). As a second sanity check, the count $12$ is between the unrestricted $24$ and the smaller choices $2$, $4$, $6$ — it lines up with "only one restriction."

Alternative: Tool #2 (Make a Systematic List): fix the driver as Bonnie. The remaining $3$ people (Ali, Carlo, Dianna) fill the other $3$ seats in $3! = 6$ ways. Repeat with Carlo as driver: another $6$ ways with the remaining (Ali, Bonnie, Dianna). Total $= 6 + 6 = 12$, again giving (D).

CCSS standards used (min grade 4)

4.OA.A.3 Solve multistep word problems using the four operations (Splitting the seating into four sequential choices ($2$, $3$, $2$, $1$) and multiplying them to get the total $2 \times 3 \times 2 \times 1 = 12$.)

⭐ Fill the strictest seat first, then multiply the leftover choices — that simple Grade 4 plan turns this AMC 8 counting problem into $2 \times 3 \times 2 \times 1 = 12$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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