2026 AIME I Question 15

Let $a,$ $b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n.$ Define an $a \times b$ *cell loop* in a $2n \times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a-2) \times (b-2)$ (possibly empty) rectangle of cells in the grid. For example, the following diagram shows a way to partition a $6 \times 6$ grid of cells into $4$ cell loops.



Find the number of ways to partition a $10 \times 10$ grid of cells into $5$ cell loops so that every cell of the grid belongs to exactly one cell loop.