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

Options

A. 210
B. 225
C. 196
D. 105

Explanation

To solve the problem of determining the number of different tickets required for travel between 20 railway stations, we can break down the solution step by step.

### Step-by-Step Solution:

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

2. **Identifying the Tickets**:
Each ticket allows travel from one station to another. If we consider the stations as A, B, C, ..., T (for a total of 20 stations), we need tickets for every possible journey between these stations.

3. **Counting the Tickets**:
- If you start from station A, you can travel to 19 other stations (B, C, D, ..., T). Thus, you need 19 tickets.
- If you start from station B, you can travel to 18 remaining stations (C, D, ..., T), since the ticket from A to B is already counted. So, you need 18 tickets.
- Continuing this pattern, if you start from station C, you will need 17 tickets, and so on.

4. **Summing the Tickets**:
- The total number of tickets required can be expressed as:
\[ 19 + 18 + 17 + ... + 1 \]
- This is the sum of the first 19 natural numbers.

5. **Using the Formula for the Sum of Natural Numbers**:
- The formula for the sum of the first \( n \) natural numbers is:
\[ S = \frac{n(n + 1)}{2} \]
- In our case, we need to sum the first 19 numbers (from 1 to 19):
\[ S = \frac{19 \times 20}{2} = 190 \]

6. **Including the Tickets for Starting from Each Station**:
- Additionally, we need to account for the ticket that allows travel from each station to itself, which is not counted in the previous sum.
- Therefore, we add 20 (one for each station) to the previous total:
\[ 190 + 20 = 210 \]

7. **Final Answer**:
- The total number of different tickets required is **210**.