CPM Homework Help : INT3 Problem 4-37

Suppose you have a $3$ by $3$ by $3$ cube. It is painted on all six faces and then cut apart into $27$ pieces, each a $1$ by $1$ by $1$ cube. If one of the cubes is chosen at random, what is the probability that:

Three sides are painted?
Hint (a):
This cube will have three sides painted. How many cubes like this are there in the figure? How many $1$ by $1$ cubes are there altogether?
Answer (a):
$P(3\text{ sides painted})=\frac{8}{27}$

Two sides are painted?
Answer (b):
$P(2\text{ sides painted})=\frac{12}{27}$

One side is painted?
Hint (c):
How many cubes have one side exposed?

No sides are painted?
Hint (d):
How many cubes have $1$ , $2$ , or $3$ sides painted? How many are left?