5×1×2

The term "5×1×2" refers to a specific arrangement of numbers used in various contexts, most notably in the field of **combinatorics** and **permutation theory**. It describes a sequence of multiplicative factors applied to a set of items, often used to calculate the number of possible arrangements or groupings.

In combinatorics, the expression can represent the number of ways to partition a set of five distinct items into groups of sizes 5, 1, and 2. This is mathematically expressed as the multinomial coefficient:

5! / (5! × 1! × 2!) = 1

However, this interpretation is less common, as it simplifies to a trivial case. More frequently, the notation is used in the context of **permutation puzzles**, particularly in the **15-puzzle** and its variants, where the arrangement of tiles follows a specific pattern.

In the **15-puzzle**, the board is typically arranged in a 4×4 grid, but the term "5×1×2" may refer to a different tiling or arrangement scheme, such as a **5-tile row, followed by a single tile, and then a 2-tile row**. This could describe a non-standard puzzle configuration or a method of encoding tile positions.

Another application appears in **group theory**, where the notation might represent a product of group elements or a decomposition of a permutation into cycles of lengths 5, 1, and 2. For example, a permutation could be written as a product of a 5-cycle, an identity (1-cycle), and a 2-cycle.

The term may also appear in **mathematical puzzles or brain teasers**, where it serves as a shorthand for a specific problem or constraint. Without additional context, its precise meaning can vary, but it generally relates to structured arrangements or decompositions involving these numerical factors.