On a railway route there are 20 stations. What is the number of different tickets required in order that it may be possible to travel from every station to every other station

Options

A. 40
B. 380
C. 400
D. 420

Answer

To solve the problem of determining the number of different tickets required for travel between 20 railway stations, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have 20 stations, and we need to find out how many different tickets are required to travel from any station to any other station.

2. **Choosing Two Stations**:
- To travel between any two stations, we need to choose 2 stations from the 20 available. The order of selection does not matter because a ticket from station A to station B is the same as a ticket from station B to station A.

3. **Using Combinations**:
- The number of ways to choose 2 stations from 20 is given by the combination formula:
\[ \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Here, \( n = 20 \) (total stations) and \( r = 2 \) (stations to choose).

4. **Calculating the Combinations**:
- Plugging in the values:
\[ \binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20!}{2! \cdot 18!} \]
- Simplifying this, we can cancel out \( 18! \):
\[ = \frac{20 \times 19}{2 \times 1} = \frac{380}{2} = 190 \]

5. **Considering Round Trips**:
- Since a ticket allows travel in both directions (from A to B and from B to A), we need to multiply the number of combinations by 2:
\[ \text{Total tickets} = 2 \times 190 = 380 \]

6. **Final Answer**:
- Therefore, the total number of different tickets required is **380**.