AMC 8 2022 Problem 23
AMC8 2022 年第 23 题
概率★★★★☆
or is placed in each of the nine squares in a 3 -by- 3 grid. Shown below is a sample configuration with three s in a line. How many configurations will have three s in a line and three s in a line?
在 3×3 方格表的九个单元格的每个中放置△或○。下图是一种有三个△在一条线上的放置方法。问在有三个△在一条线上，并且有三个○在一条线上的放置方法有多少种？
- A.39
- B.42
- C.78
- D.84
- E.96
正确答案 · Correct Answer
D
解析 · Solution
解法1
常规解法 · 行/列两类，各自去重
△线与○线不能相交（交点不能既△又○），故二者要么同为行、要么同为列。都用行：选△行 3 × 选○行 2 × 第三行随意 ；但「三行全单色」的 6 个配置各被数了 2 次（如 △△△/△△△/○○○ 既算「△行=第1行」又算「△行=第2行」），去重减 6 得 42。都用列同理 42。两类互斥（有△行又有○行时每列都混色），合计 42+42=84。
巧解 · 对行做容斥，再 ×2
「有△行且有○行」的配置数：每行 8 种花样，（总 − 无△行 − 无○行 + 都无）。列同理 42，且两类互斥（有△行+○行 ⇒ 每列必混色），合计 84。
易错：直接 48+48=96 会把 6+6 个「全单色」配置各重复一次。
Solution 1
Elementary · rows vs columns, de-duplicate each
The all- line and the all- line cannot cross, so both are rows or both are columns. Both rows: naively 3 (choose the row) (choose the row) (third row free) =48; but the 6 configurations in which all three rows are single-symbol are each counted twice, so subtract 6 to get 42. Both columns gives 42 by symmetry. The two cases are disjoint (a row together with an row forces every column to be mixed). Total 42+42=84.
Slick · inclusion–exclusion on rows, then ×2
The number of grids having both a row and an row is, with 8 patterns per row, (all, minus no--row, minus no--row, plus neither). Columns give 42 likewise, and the two cases are disjoint, so the total is 84.
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