To solve the problem of finding the number of ways to color a 3×3 grid such that each row and column contains exactly one of each color (a Latin square of order 3), we can follow these steps:

Key Observations:

A Latin square of order 3 requires each row and column to have all three distinct colors (no repeats).

1. First Row: The first row can be any permutation of the 3 colors. There are (3! = 6) possible permutations.
2. Second Row: For each first row, the second row must be a derangement of the first row (no color in the same column as the first row). For 3 elements, the number of derangements is (D(3) = 2) (since derangements of [A,B,C] are [B,C,A] and [C,A,B]).
3. Third Row: Once the first two rows are fixed, the third row is uniquely determined (each column must have the missing color).

Total Number of Ways:

Multiply the number of permutations for the first row by the number of derangements for the second row:
(6 \times 2 = 12).

Answer: (\boxed{12})

作者声明:本文包含人工智能生成内容。