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.

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