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
📏 Medium solution 💡 3 insights
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Problem
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?

Pick an answer.

(A)
4
(B)
5
(C)
6
(D)
8
(E)
10

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

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.

1STEP 1

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.

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

Each route is 3 E's and 2 N's in some order, so list them systematically — there are exactly 10 such routes.

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

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.

Bad starts: routes beginning with EN… or NE….nFrom the list: #2 NE…, #3 NE…, #4 NE…, #5 EN…, #6 EN…, #7 EN… — that's 6 bad routes.
4STEP 4

Cross out those 6 from the 10, and only NNEEE, EENNE, EENEN, EEENN survive — 4 safe routes.

Safe routes: {NNEEE, EENNE, EENEN, EEENN} → count = 4.
5STEP 5

Answer: 4 safe routes — choice (A).

Number of safe routes = 10 - 6 = 4 → (A)
Answer
4
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.
💡Key takeaway

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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Same archetype — closest grade level first.