Solution to Homework 4, Math 152-205, additional problem:

The homogeneous form of the system either has a unique solution
(namely all variables equal 0) or infinitely many.  Accordingly it is
impossible for both (2) (i.e., a unique solution for some value of c)
and (3) (i.e., infinitely many solutions for some value of c) to occur.

Aside from that, everything is possible, keeping in mind that at least
one of (1) (i.e., no solutions), (2), (3) must occur:

Only (1): Two equations: x1 = 5,  x1 = 6.
(1) and (2): Five equations: x1 = 0, x2 = 0, x3 = 0, x4 = 0, x1 = c.
(1) and (3): Two equations: x1 = 5, x1 = 2 c.
Only (2): Four equations: x1 = 5, x2 = 0, x3 = 0, x4 = 0.
Only (3): One equation: x1 = 5.

If you want the parameter  c  to explicitly appear in all of these systems,
then replace all the 5's by c's, and the 6 by c+1.