What are dimensions of a physical quantity?

If the dimensions of a physical quantity are given by M^(x) L^(y) T^(z) , then physical quantity may be

If the dimension of a physical quantity are given by M^a L^b T^c, then the physical quantity will be

If the dimension of a physical quantity are given by M^a L^b T^c, then the physical quantity will be

If the dimension of a physical quantity are given by M^a L^b T^c, then the physical quantity will be

If the dimension of a physical quantity are given by M^a L^b T^c, then the physical quantity will be

In the dimension of a physical quantities are given by M^(0)L^(1)T^(0) , then the physical quantity will be

Find the dimensions of physical quantity X in the equation "Force" = ( X) /("Density") .

Give an example of (a) a physical quantiy which has a unit but no dimensions. (b) a physical quantity which has neither unit no dimensions. (c ) a constant which has a unit. (d) a constant which has no unit.

As far as mechanics is concerned all systems of units are having the same dimensions for a given physical quantity. But for thermodynamics, electricity, magnetism and electromagnetics this is not the case. Why?

In Column-I, some physical quantities are given and in column-II, dimensions of the physical quantities are given . Match the entries in column-I to each entry in Column-II.