To determine the number of shift-reduce conflicts in the $LR(0)$ ACTION table for the given grammar, we first need to construct the $LR(0)$ items and the corresponding state machine. Let's go through this process step by step.

1. Start with the augmented grammar:
   • $S' \rightarrow S$ (where $S'$ is the new start symbol)
   • $S \rightarrow ACB$
   • $A \rightarrow aA \mid \epsilon$
   • $C \rightarrow cC \mid \epsilon$
   • $B \rightarrow bB \mid b$
2. Construct the $LR(0)$ items:
   • Begin with the closure of the initial item: $[S' \rightarrow \cdot S]$
   • Add items for the successive derivations and their corresponding closure steps.
3. Develop the canonical collection of $LR(0)$ items (states) and define the transitions:
   • Each state reflects a possible configuration of the grammar as you parse left-to-right.
   • Each transition corresponds to shifting a symbol.
4. Create the ACTION table:
   • Entries are defined based on whether a state is complete (has a production ending in a dot) or can shift.
   • Analyze each state to determine if it leads to a shift or a reduce, or possibly both (indicating a conflict).

In constructing the $LR(0)$ parsing table, conflicts typically arise in states where:

• A production rule can be completed (reduce action) and a shift is possible based on the current conditions (lookahead).

For this specific grammar:

• Each non-terminal introduces potential epsilon transitions, increasing the likelihood of conflicts.

Analyzing the shift-reduce dynamics for this grammar, you obtain a total of 5 shift-reduce conflicts in the ACTION table of the $LR(0)$ parser.

Conclusion: The number of shift-reduce conflicts that occur in the $LR(0)$ ACTION table is \(5\), making the correct option 5.