AMC 8 · 2014 · #11

Easy mode Grade 7

Problem

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?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

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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