Question

There are 10 stations on a railway line. The number of different journey tickets that are required by the authorities is

• A. 10
• B. 90
• C. 81
• D. 10

Answer 1

The Correct Option is B

To determine the number of different journey tickets required, consider that a passenger can travel between any two distinct stations. Let's label the stations as \( S_1, S_2, \ldots, S_{10} \).

For each pair of distinct stations \( S_i \) and \( S_j \) where \( i \neq j \), one journey ticket is required. The number of ways to choose 2 stations out of 10 is given by the combination formula:

$$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$

Here, \( n = 10 \) and \( r = 2 \). Plugging these into the formula:

$$ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 $$

Each selected pair of stations corresponds to two different journey tickets: one for each direction (from \( S_i \) to \( S_j \) and from \( S_j \) to \( S_i \)). Therefore, the total number of tickets needed is twice the number of ways to choose 2 stations:

$$ 2 \times 45 = 90 $$

Thus, the number of different journey tickets required by the authorities is 90.