To have an average of 5, the sum of the three must be 15.

Here is a fact about positive numbers we can use:

If two positive numbers have a certain sum, their
product is largest when they are closest together
and smallest when they are farthest apart.

However, here we have three integers with that sum, not 
just two.  So that complicates matters
a bit.

But, we can consider only two numbers and use that
fact above. Here's how:

Suppose the three integers are a, b, and c, where 

a < b < c.  We can consider just two numbers,

(a+b) and c.

We want to choose (a+b) and c as far apart as possible.

To do that we choose (a+b) as small as possible, which is 3,
when a=1 and b=2, and that choice will make c as much larger 
than 3 as possible.  In order to have sum 15, c must be 12,
and 12 is as far away from 3 as we can get.  So the smallest 
possible product is 1*2*12 = 24.

Edwin