How many different capacitances can you produce using three capacitors if you have all three connected in the circuit each time?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To determine how many different capacitances can be produced using three capacitors connected in the circuit each time, we can analyze the possible configurations of these capacitors. Let's denote the three capacitors as C1, C2, and C3.

### Step-by-Step Solution:

1. **Understanding Connections**: Each capacitor can be connected in two ways: either in series (S) or in parallel (P). Since we have three capacitors, we need to consider all possible combinations of these connections.

2. **Combination of Connections**: For three capacitors, each capacitor has two connection options (S or P). Therefore, the total number of combinations can be calculated as:
\[ \text{Total combinations} = 2^n \]
where \( n \) is the number of capacitors. Here, \( n = 3 \):
\[ \text{Total combinations} = 2^3 = 8 \]

3. **Listing the Combinations**: Now, let's list the different combinations of connections:
- **Combination 1**: All three capacitors in series (C1, C2, C3 in series).
- **Combination 2**: C1 and C2 in series, C3 in parallel with them.
- **Combination 3**: C1 and C3 in series, C2 in parallel with them.
- **Combination 4**: C2 and C3 in series, C1 in parallel with them.
- **Combination 5**: C1 and C2 in parallel, C3 in series with them.
- **Combination 6**: C1 and C3 in parallel, C2 in series with them.
- **Combination 7**: C2 and C3 in parallel, C1 in series with them.
- **Combination 8**: All three capacitors in parallel (C1, C2, C3 in parallel).

4. **Conclusion**: Therefore, the total number of different capacitances that can be produced using three capacitors, with each combination of series and parallel connections, is **8**.