Step 1: Concept
Each coin can either be selected or not selected (2 choices).
Step 2: Combinations
For 5 coins, total combinations $= 2^5 = 32$.
Step 3: Constraint
We must exclude the case where no coins are selected (sum = 0).
Step 4: Conclusion
Total sums $= 32 - 1 = 31$.
Final Answer: (C)