AMC 8 · 2014 · #11

Grade 7 counting

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

combinations-basiclattice-pathscomplementary-counting complementary-countingidentify-subproblems ↑ Prerequisites: combinations-basicsystematic-enumeration

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Problem

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2014 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: Jack bikes from his house to Jill's house, $3$ blocks east and $2$ blocks north away. Every block he picks east or north (no backtracking), and the whole trip is $5$ blocks. One corner — $1$ block east and $1$ block north of Jack's house — is dangerous and must be avoided. How many different $5$-block routes work?

Givens: Jack's house at $(0,0)$, Jill's house at $(3,2)$; Each block Jack moves either east (E) or north (N) — never west or south; Total trip is exactly $3 + 2 = 5$ blocks; Forbidden corner: $(1,1)$, one block east and one block north of Jack; Answer choices: (A) $4$, (B) $5$, (C) $6$, (D) $8$, (E) $10$

Unknowns: The number of $5$-block east/north routes from $(0,0)$ to $(3,2)$ that do not pass through $(1,1)$

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #1 Draw a Diagram, #3 Eliminate Possibilities, #16 Change Focus / Count the Complement

Only $10$ total routes exist, so we don't need any combinatorics formula — Tool #2 (Systematic List) can write them all down. Tool #1 (Draw a Diagram) gives us a $3 \times 2$ grid to read each route off of, and Tool #3 (Eliminate Possibilities) crosses out the ones that hit $(1,1)$. Tool #16 (Count the Complement) is the natural review check: instead of listing safe routes, list bad routes and subtract from $10$.

Execute — Answer: A

#1 Draw a Diagram 5.G.A.1 Step 1

Draw a $3 \times 2$ grid of streets with Jack's house at the bottom-left corner $(0,0)$ and Jill's house at the top-right corner $(3,2)$.
Mark the dangerous corner at $(1,1)$ with an X.
Every route is a staircase of E (east, right) and N (north, up) moves from $(0,0)$ to $(3,2)$.

Grid: $4$ columns ($x = 0,1,2,3$) by $3$ rows ($y = 0,1,2$). Forbidden point: $(1,1)$.

💡 Putting houses and the dangerous corner on a coordinate grid turns a street-direction word problem into a picture of dots you can point to.

#2 Make a Systematic List 7.SP.C.8 Step 2

Each route uses $3$ E's and $2$ N's in some order, so each route is a $5$-letter string.
List every such string in alphabetical order (E before N).
There are $\binom{5}{2} = 10$ of them, but we don't need that formula — we can simply enumerate.
Order: pick the positions of the two N's from left to right.

All $10$ routes (positions of the two N's shown in brackets):\n$1)\ \text{NNEEE}\ [1,2]$\n$2)\ \text{NENEE}\ [1,3]$\n$3)\ \text{NEENE}\ [1,4]$\n$4)\ \text{NEEEN}\ [1,5]$\n$5)\ \text{ENNEE}\ [2,3]$\n$6)\ \text{ENENE}\ [2,4]$\n$7)\ \text{ENEEN}\ [2,5]$\n$8)\ \text{EENNE}\ [3,4]$\n$9)\ \text{EENEN}\ [3,5]$\n$10)\ \text{EEENN}\ [4,5]$

💡 Choosing a fixed ordering rule (sort by where the N's appear) guarantees we hit every route exactly once.

#3 Eliminate Possibilities 4.OA.A.3 Step 3

A route reaches $(1,1)$ exactly when, somewhere along the way, it has done $1$ E and $1$ N.
That happens when the first two moves are one E and one N — i.e., the route starts with EN or NE.
Trace the prefix of each listed route: if the first two letters are $\{E,N\}$ in either order, the route visits $(1,1)$.

Bad starts: routes beginning with $\text{EN}\ldots$ or $\text{NE}\ldots$.\nFrom the list: $\#2\ \text{NE}\ldots,\ \#3\ \text{NE}\ldots,\ \#4\ \text{NE}\ldots,\ \#5\ \text{EN}\ldots,\ \#6\ \text{EN}\ldots,\ \#7\ \text{EN}\ldots$ — that's $6$ bad routes.

💡 Reaching $(1,1)$ after exactly two moves is the only way a $5$-move E/N path can hit it, so we just check the first two letters.

#3 Eliminate Possibilities 4.OA.A.3 Step 4

Cross out those $6$ bad routes from the list of $10$.
What remains is the set of safe routes.
They are: $\#1$ NNEEE (goes up first, well above $(1,1)$), $\#8$ EENNE, $\#9$ EENEN, $\#10$ EEENN (all start with EE, so they pass $(2,0)$ before any north move and miss $(1,1)$).

Safe routes: $\{\text{NNEEE},\ \text{EENNE},\ \text{EENEN},\ \text{EEENN}\}$ $\Rightarrow$ count $= 4$.

💡 After the elimination, just count what's left — no formula needed.

#2 Make a Systematic List 4.OA.A.3 Step 5

Answer: $4$ safe routes, which is choice (A).

Number of safe routes $= 10 - 6 = 4 \;\Rightarrow\; \textbf{(A)}$

💡 The systematic list directly gives the final count.

[1] #1 5.G.A.1 Draw a $3 \times 2$ grid of streets with Jack's house at the bottom-left corner

[2] #2 7.SP.C.8 Each route uses $3$ E's and $2$ N's in some order, so each route is a $5$-letter

[3] #3 4.OA.A.3 A route reaches $(1,1)$ exactly when, somewhere along the way, it has done $1$ E

[4] #3 4.OA.A.3 Cross out those $6$ bad routes from the list of $10$. What remains is the set of

[5] #2 4.OA.A.3 Answer: $4$ safe routes, which is choice (A).

Review

Reasonableness: The forbidden corner $(1,1)$ sits very near the start, so it blocks many routes — $6$ out of $10$. Only routes that either go straight up first (NN...) or go two steps east first (EE...) sneak around it, and there are clearly only $4$ such routes. The count $4$ matches choice (A) and feels right for such a small grid.

Alternative: Tool #16 (Count the Complement): instead of listing safe routes, count bad ones using the grid. Routes from $(0,0)$ to $(1,1)$: $2$ (EN or NE). Routes from $(1,1)$ to $(3,2)$: $3$ (EEN, ENE, NEE). By the multiplication principle, bad routes $= 2 \times 3 = 6$. Total routes $= 10$. Safe $= 10 - 6 = 4$. Same answer.

CCSS standards used (min grade 7)

5.G.A.1 Use a pair of perpendicular number lines (coordinate system) to locate points (Placing Jack's house at $(0,0)$, Jill's house at $(3,2)$, and the dangerous corner at $(1,1)$ on a coordinate grid so the problem becomes a picture.)
4.OA.A.3 Solve multistep word problems posed with whole numbers (Tracking the running count of east and north moves along each route and checking the constraint "never reach $(1,1)$".)
7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation (Generating the sample space of all $10$ possible E/N routes as an organized list, in alphabetical order, with no duplicates and none missing.)

⭐ When the total number of options is small (here, just $10$), you don't need a counting formula — a Grade 7-style organized list of every route, then crossing out the bad ones, gets the answer.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 7)

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