If five seats are vacant in a train, the number of ways three passengers can sit is:

Options

A. (a) `10`
B. (b) `20`
C. (c) `30`
D. (d) `60`

Answer

To solve the problem of how many ways three passengers can sit in five vacant seats in a train, we can use the concept of permutations since the order in which the passengers sit matters.

### Step-by-Step Solution:

1. **Identify the Total Seats and Passengers**:
We have a total of 5 vacant seats and we need to arrange 3 passengers in these seats.

2. **Define the Variables**:
Let \( N \) be the total number of seats (5) and \( R \) be the number of passengers (3).

3. **Use the Permutation Formula**:
The formula for permutations is given by:
\[ P(N, R) = \frac{N!}{(N-R)!} \]
Here, \( N = 5 \) and \( R = 3 \).

4. **Substitute the Values into the Formula**:
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} \]

5. **Calculate Factorials**:
- First, calculate \( 5! \):
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
- Next, calculate \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

6. **Plug in the Factorials**:
\[ P(5, 3) = \frac{120}{2} = 60 \]

7. **Conclusion**:
The number of ways three passengers can sit in five seats is 60. Therefore, the correct answer is option D, which is 60.