If three wires of equal resistance are given then number of combination they cany be made to give different resistance is

Options

A. 4
B. 3
C. 5
D. 2

Explanation

To solve the question of how many different combinations of three wires of equal resistance can be made to give different resistances, we can follow these steps:

### Step 1: Understand the configurations
We can connect resistors in two basic configurations: series and parallel. We will explore all possible combinations of these configurations using three resistors.

### Step 2: Series Connection
When all three resistors (let's denote the resistance of each wire as \( R \)) are connected in series, the total resistance \( R_s \) is given by:
\[ R_s = R + R + R = 3R \]
This is our first combination.

### Step 3: Parallel Connection
When all three resistors are connected in parallel, the total resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} \implies R_p = \frac{R}{3} \]
This is our second combination.

### Step 4: Two in Parallel, One in Series
Now, let's connect two resistors in parallel and the third resistor in series with this combination. The equivalent resistance of the two resistors in parallel is:
\[ \frac{1}{R_{parallel}} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \implies R_{parallel} = \frac{R}{2} \]
Now, adding the third resistor in series gives:
\[ R_{total} = R_{parallel} + R = \frac{R}{2} + R = \frac{3R}{2} \]
This is our third combination.

### Step 5: Two in Series, One in Parallel
Next, we can connect two resistors in series and then connect this combination in parallel with the third resistor. The equivalent resistance of the two resistors in series is:
\[ R_{series} = R + R = 2R \]
Now, connecting this in parallel with the third resistor gives:
\[ \frac{1}{R_{total}} = \frac{1}{2R} + \frac{1}{R} = \frac{1 + 2}{2R} = \frac{3}{2R} \implies R_{total} = \frac{2R}{3} \]
This is our fourth combination.

### Step 6: Count the Combinations
We have found four distinct combinations of resistances:
1. \( 3R \) (all in series)
2. \( \frac{R}{3} \) (all in parallel)
3. \( \frac{3R}{2} \) (two in parallel, one in series)
4. \( \frac{2R}{3} \) (two in series, one in parallel)

### Conclusion
Thus, the total number of different combinations of three wires of equal resistance that can be made is **4**.