# Possible Partitions

The integers from 1 through 10 (inclusive) are divided into three groups, each containing at least one number. These groups satisfy the additional property that if $$x$$ is in a group and $$2x \leq 10$$, then $$2x$$ is in the same group. How many different ways are there to create the groups?

Details and assumptions

2 ways are considered different, if we are unable to match up the groups. For example, the way $$A= \{ 1, 2, 4, 8\}$$, $$B = \{3, 5, 6, 10\}$$, $$C=\{7,9\}$$, will be considered the same way as the grouping $$X = \{7,9\}$$ , $$Y=\{10, 6, 5, 3\}$$, $$Z=\{8, 4, 2, 1\}$$.

Each integer appears in exactly one of the groups.

×