AMC 10 · 2007 · #20

Grade 7 geometry-2d
Archetype Branching State Enumeration └ Sub-archetype Ordered Placement Count
combinations-basicpermutations-basicsystematic-enumeration identify-subproblems ↑ Prerequisites: combinations-basic
📏 Medium solution 💡 3 insights
Problem
25 square blocks are arranged in a 5 × 5 square grid. How many different groups of 3 blocks can be selected so that no two selected blocks lie in the same row or the same column?

Pick an answer.

(A)
100
(B)
125
(C)
600
(D)
2300
(E)
3600

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The rule "no two blocks share a row or column" means the 3 chosen blocks sit in 3 distinct rows and 3 distinct columns. That lets Tool #7 (Identify Subproblems) split the count into three independent decisions: (1) which 3 of the 5 rows to use, (2) which 3 of the 5 columns to use, and (3) how to pair those columns with those rows. Tool #2 (Make a Systematic List) handles each small count, and Tool #16 (Change Focus) is the key reframe — instead of hunting for legal block-triples directly, we choose rows, choose columns, then match them.

1STEP 1

Reframe as rows, columns, matching

Name a block by its (row, column). The 3 blocks use 3 different rows and 3 different columns — so pick rows, pick columns, then pair them.

group of 3 blocks = (3 rows) × (3 columns) × (how to pair them)
2STEP 2

Count ways to pick the rows

Choosing 3 of the 5 rows is the same as leaving 2 out, and there are 10 such pairs: 10 ways for rows, 10 for columns.

C(5, 3) = (5 · 4)/2 = 10 ways for rows, C(5, 3) = 10 ways for columns
3STEP 3

Count ways to match them up

With the rows and columns fixed, give each chosen column a row: 3 × 2 × 1 = 6 matchings, each a different legal trio.

3! = 3 × 2 × 1 = 6 matchings
4STEP 4

Multiply the three counts

The three choices are independent, so multiply: 10 × 10 × 6 = 600 — every valid trio counted exactly once.

10 × 10 × 6 = 600 → (C)
Answer
600
The answer 600 must beat the plain (rule-free) count of C(25, 3) = 2300 picks of 3 blocks, and it does — the row/column rule throws many groups out, so a number well below 2300 is expected, ruling out (D) 2300 and (E) 3600. A second, independent count confirms 600: place the blocks one at a time. The first block has 25 choices; it blocks its whole row and column, leaving 16 legal cells for the second; the second block leaves 9 legal cells for the third. That is 25 × 16 × 9 = 3600 ordered ways, but the group is unordered, so divide by 3! = 6: 3600 / 6 = 600. Both methods give 600, matching (C).
💡Key takeaway

When picks can't share a row or column, choose the rows, choose the columns, then count how to line them up — and multiply the three counts together.

  • Reframe as rows, columns, matching
  • Count ways to pick the rows
  • Count ways to match them up
  • Multiply the three counts

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