To determine which of the given cases has the maximum resistance using three resistors, we will analyze each configuration step by step.
### Step 1: Understanding the configurations
We have four configurations to consider:
1. All three resistors in series.
2. All three resistors in parallel.
3. Two resistors in series and one in parallel.
4. Two resistors in parallel and one in series.
### Step 2: Calculating resistance for each configuration
Assuming each resistor has a resistance of \( R = 1 \, \Omega \).
#### Case 1: All resistors in series
The total resistance \( R_s \) in series is given by:
\[
R_s = R_1 + R_2 + R_3 = 1 + 1 + 1 = 3 \, \Omega
\]
#### Case 2: All resistors in parallel
The total resistance \( R_p \) in parallel is given by:
\[
\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 3
\]
Thus,
\[
R_p = \frac{1}{3} \, \Omega
\]
#### Case 3: Two resistors in series and one in parallel
Let’s consider \( R_1 \) and \( R_2 \) in series and \( R_3 \) in parallel:
\[
R_{s} = R_1 + R_2 = 1 + 1 = 2 \, \Omega
\]
Now, this series combination is in parallel with \( R_3 \):
\[
\frac{1}{R_{total}} = \frac{1}{R_s} + \frac{1}{R_3} = \frac{1}{2} + \frac{1}{1} = \frac{1 + 2}{2} = \frac{3}{2}
\]
Thus,
\[
R_{total} = \frac{2}{3} \, \Omega
\]
#### Case 4: Two resistors in parallel and one in series
Let’s consider \( R_1 \) and \( R_2 \) in parallel and \( R_3 \) in series:
\[
\frac{1}{R_{p}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{1} + \frac{1}{1} = 2
\]
Thus,
\[
R_{p} = \frac{1}{2} \, \Omega
\]
Now, this parallel combination is in series with \( R_3 \):
\[
R_{total} = R_{p} + R_3 = \frac{1}{2} + 1 = \frac{3}{2} \, \Omega
\]
### Step 3: Summary of resistances
Now we summarize the resistances calculated:
1. Series: \( R_s = 3 \, \Omega \)
2. Parallel: \( R_p = \frac{1}{3} \, \Omega \)
3. Two in series, one in parallel: \( R_{total} = \frac{2}{3} \, \Omega \)
4. Two in parallel, one in series: \( R_{total} = \frac{3}{2} \, \Omega \)
### Step 4: Conclusion
The maximum resistance is found in the series configuration:
\[
\text{Maximum Resistance} = 3 \, \Omega
\]
