Given three equal resistors, how many different combination of all the three resistor can be made

Options

A. Six
B. Five
C. Four
D. Three

Explanation

To determine how many different combinations can be made with three equal resistors, we can analyze the possible configurations systematically.

### Step-by-Step Solution:

1. **Combination 1: All Resistors in Series**
- Connect all three resistors (R1, R2, R3) in series.
- The total resistance (R_total) is given by:
\[ R_{\text{total}} = R_1 + R_2 + R_3 = 3R \]

2. **Combination 2: All Resistors in Parallel**
- Connect all three resistors in parallel.
- The total resistance (R_total) is given by:
\[ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} = \frac{3}{R} \implies R_{\text{total}} = \frac{R}{3} \]

3. **Combination 3: Two Resistors in Series and One in Parallel**
- Connect two resistors in series and the third one in parallel with this combination.
- Let’s say R1 and R2 are in series, and R3 is in parallel with them.
- The resistance of R1 and R2 in series:
\[ R_{12} = R_1 + R_2 = 2R \]
- Now, R3 in parallel with R12:
\[ \frac{1}{R_{\text{total}}} = \frac{1}{R_{12}} + \frac{1}{R_3} = \frac{1}{2R} + \frac{1}{R} = \frac{3}{2R} \implies R_{\text{total}} = \frac{2R}{3} \]

4. **Combination 4: Two Resistors in Parallel and One in Series**
- Connect two resistors in parallel and the third one in series with this combination.
- Let’s say R1 and R2 are in parallel, and R3 is in series with them.
- The resistance of R1 and R2 in parallel:
\[ \frac{1}{R_{12}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{2}{R} \implies R_{12} = \frac{R}{2} \]
- Now, R3 in series with R12:
\[ R_{\text{total}} = R_{12} + R_3 = \frac{R}{2} + R = \frac{3R}{2} \]

### Summary of Combinations:
1. All in Series: \( R_{\text{total}} = 3R \)
2. All in Parallel: \( R_{\text{total}} = \frac{R}{3} \)
3. Two in Series, One in Parallel: \( R_{\text{total}} = \frac{2R}{3} \)
4. Two in Parallel, One in Series: \( R_{\text{total}} = \frac{3R}{2} \)

### Conclusion:
Thus, there are **four different combinations** of the three equal resistors.