Rajdhani express going from Bombay to Delhi stops at 5 intermediate stations. 10 passengers enter the train during the journey with ten different ticket of two classes .The number of different sets of tickets they may have is

Options

A. `""^(15)C_(10)`
B. `""^(20)C_(10)`
C. `""^(30)C_(10)`
D. None of these

Answer

To solve the problem of how many different sets of tickets the 10 passengers may have on the Rajdhani Express, we can break down the solution into several steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Stations**:
The Rajdhani Express stops at 5 intermediate stations between Bombay and Delhi.

2. **Determine the Ticket Options**:
Each ticket allows a passenger to travel from one station to another. For each pair of stations, a ticket can be purchased. The number of ways to choose two stations from the 5 intermediate stations can be calculated using combinations.

3. **Calculate the Number of Tickets for One Class**:
The number of ways to choose 2 stations from 5 is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
Here, \( n = 5 \) (the intermediate stations) and \( r = 2 \) (the stations to choose).
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
Therefore, there are 10 different tickets for one class.

4. **Consider Both Classes**:
Since there are 2 classes of tickets, the total number of different tickets available is:
\[ \text{Total tickets} = 10 \text{ (Class 1)} + 10 \text{ (Class 2)} = 20 \]

5. **Calculate the Total Number of Different Sets of Tickets**:
Now, we need to determine how many different sets of tickets can be assigned to 10 passengers. Each passenger can choose any of the 20 tickets. Therefore, the total number of different sets of tickets that can be assigned to the 10 passengers is given by:
\[ \text{Total sets of tickets} = 20^{10} \]

6. **Final Answer**:
The total number of different sets of tickets that the 10 passengers may have is \( 20^{10} \).