Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #11
Grade 7 countingPick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 × 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.
Draw the street grid
Put Jack at (0,0) and Jill at (3,2) on a grid, marking danger corner (1,1) with an X. Each route is a staircase of east and north moves.
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.
5.G.A.1Draw A DiagramList all ten routes
Each route is 3 E's and 2 N's in some order, so list them systematically — there are exactly 10 such routes.
Choosing a fixed ordering rule (sort by where the N's appear) guarantees we hit every route exactly once.
7.SP.C.8Make A Systematic ListSpot the routes through the corner
A route hits (1,1) only if its first two moves are one E and one N — starting EN or NE — which knocks out 6 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.
4.OA.A.3Eliminate PossibilitiesCross out the bad routes
Cross out those 6 from the 10, and only NNEEE, EENNE, EENEN, EEENN survive — 4 safe routes.
After the elimination, just count what's left — no formula needed.
Crossing the routes that pass through the dangerous corner out of the full set of routes leaves exactly the safe routes, so the number of safe routes equals the total number of routes minus the number that pass through the corner.
▸ Why?
Every route either passes through the dangerous corner or avoids it, and none can do both, so the corner-passing routes and the safe routes together make up all the routes with no route left out and none counted twice.
▸ Why?
Because the safe routes and the corner-passing routes together fill out the whole set of routes, taking the corner-passing ones away from that whole leaves just the safe ones, since removing undoes combining.
Count the safe routes
Answer: 4 safe routes — choice (A).
The systematic list directly gives the final count.
4.OA.A.3Make A Systematic ListWhen 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.
- Draw the street grid
- List all ten routes
- Spot the routes through the corner
- Cross out the bad routes
- Count the safe routes
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