There are 12 steamers plying between A and B. In how many ways could the round trip from A be made if the return was made on (i) the same steamer? (ii) a different steamer?

Explanation

To solve the problem of how many ways a round trip can be made from A to B and back, we will break it down into two parts based on the conditions provided.

### Part (i): Return on the Same Steamer

1. **Choose a Steamer for the Outward Journey**:
- There are 12 steamers available to choose from for the journey from A to B.
- Thus, the number of ways to choose a steamer for the outward journey is **12**.

2. **Return on the Same Steamer**:
- Since the condition states that the return must be made on the same steamer, there is only **1 way** to return (the same steamer chosen for the outward journey).

3. **Total Ways for Part (i)**:
- By the fundamental principle of multiplication, the total number of ways to make the round trip is:
\[ \text{Total Ways} = \text{Ways to go} \times \text{Ways to return} = 12 \times 1 = 12 \]

### Part (ii): Return on a Different Steamer

1. **Choose a Steamer for the Outward Journey**:
- Again, there are 12 steamers available for the journey from A to B.
- The number of ways to choose a steamer for the outward journey is **12**.

2. **Choose a Different Steamer for the Return Journey**:
- Since the return must be made on a different steamer, after choosing one steamer for the outward journey, there are **11 steamers left** to choose from for the return journey.

3. **Total Ways for Part (ii)**:
- Using the fundamental principle of multiplication again, the total number of ways to make the round trip is:
\[ \text{Total Ways} = \text{Ways to go} \times \text{Ways to return} = 12 \times 11 = 132 \]

### Final Answers:
- (i) The number of ways to make the round trip on the same steamer is **12**.
- (ii) The number of ways to make the round trip on a different steamer is **132**.