The maximum number of co-existing phases in a C-system is C-1.

Explanation:
A C-system refers to a system that contains multiple components. Each component can exist in different phases, such as solid, liquid, or gas. The maximum number of co-existing phases in a C-system can be determined using the Gibbs phase rule.

The Gibbs phase rule states that for a system with C components and P phases, the number of degrees of freedom (F) is given by:

F = C - P + 2

In this equation, F represents the number of independent variables that can be varied without changing the number of phases.

To determine the maximum number of co-existing phases, we need to consider the condition where F is equal to zero. This means that there are no independent variables and the system is at equilibrium.

Setting F = 0 in the Gibbs phase rule equation, we have:

0 = C - P + 2

Rearranging the equation, we get:

P = C + 2

Therefore, the maximum number of co-existing phases in a C-system is C + 2.

However, we need to subtract 1 from the result because one phase is always a reference phase, usually the standard state of the pure component. This reference phase is necessary to define the properties and behavior of the other phases.

Hence, the correct answer is option (b) C-1, which represents the maximum number of co-existing phases in a C-system.