Gauss's Law problem

A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of lambda, and the cylinder has a net charge per unit length of 2lambda. From this information, use Gauss's law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance r from the axis.

So, I am done with part A. My question on B is: I found that in the solution we use 3 lambda as the charge per unit length. Why do we not use 2 lambda? because don't we consider the charge INSIDE Gauss's surface? which is outside of the ENTIRE cylinder? So why are we considering just the one on the outer surface of the cylinder?

Only the charge on the outer surface of the cylinder is included. When constructing a Gaussian surface to determine electric field, we use a Gaussian "hollow" cylindrical surface coaxial with the wire and metal cylinder. The inner Gaussian surface is placed within the walls of the metal cylinder, and the outer Gaussian surface is a distance r from the axis. Because the inner Gaussian surface is in a conductive medium, there is no electric field. The only electric field normal to a Gaussian surface is at the outer surface. The total charge enclosed by the Gaussian cylinder is that which resides on the outer surface of the metal cylinder.