Microsoft SQL Server Analysis Services Function Page
Function Name:	Count (Dimension)
Category:	Metadata
Description:	The Count function, when applied to dimensions, returns the number of hierarchies in a cube.
Syntax:	DIMENSIONS.COUNT
Technet Link:	MDX Count (Dimension) Function

The query below shows there are a total of 208 hierarchies in the Adventure Works cube.   WITH       MEMBER [Measures].[NumberOfDimensions] AS       Dimensions.Count SELECT       [Measures].[NumberOfDimensions] ON COLUMNS       FROM       [Adventure Works]

The query below shows there are a total of 140 hierarchies in the Mined Customers cube.   WITH       MEMBER [Measures].[NumberOfDimensions] AS       Dimensions.Count SELECT       [Measures].[NumberOfDimensions] ON COLUMNS       FROM       [Mined Customers]