AMC 8 · 2014 · #11
Easy mode Grade 7Problem
Picture a grid of streets. Jack lives at one corner. Jill lives 3 blocks east and 2 blocks north of Jack.
Jack bikes to Jill's house. At every corner he can only go east or north — never south or west. The whole trip takes exactly 5 blocks.
There is one corner Jack must stay away from. It is the corner 1 block east and 1 block north of his house. He cannot pass through it.
How many different paths can Jack take from his house to Jill's house?
Pick an answer.
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.
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
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) $10$
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
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 $(3,2)$.
💡 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.
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.
💡 Choosing a fixed ordering rule (sort by where the N's appear) guarantees we hit every route exactly once.
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 $(1,1)$.
💡 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.
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 $(1,1)$).
💡 After the elimination, just count what's left — no formula needed.
4.OA.A.3 Step 5 Answer: $4$ safe routes, which is choice (A).
💡 The systematic list directly gives the final count.
5.G.A.1 Draw a $3 \times 2$ grid of streets with Jack's house at the bottom-left corner 7.SP.C.8 Each route uses $3$ E's and $2$ N's in some order, so each route is a $5$-letter 4.OA.A.3 A route reaches $(1,1)$ exactly when, somewhere along the way, it has done $1$ E 4.OA.A.3 Cross out those $6$ bad routes from the list of $10$. What remains is the set of 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.1Use 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.3Solve 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.8Find 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.
⭐ 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.
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