Question

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

• A. 92
• B. 90
• C. 91
• D. None of these

Hint:

Think of each ticket as an ordered choice of a start and end station: use permutations directly, or combinations doubled for direction.

Answer 1

The Correct Option is B

Step 1: Set up the count needed.
A ticket is bought for one direction between two stations, so a ticket from station A to station B is different from a ticket from B to A. With 10 stations, we first pick any 2 stations out of 10, then account for both directions between that pair.

Step 2: Count ways to pick 2 stations.
The number of ways to choose 2 stations out of 10, where order does not matter yet, is \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \). This counts each pair of stations once.

Step 3: Account for direction.
Every pair of stations needs two tickets, one for each direction, so multiply by 2: \( 45 \times 2 = 90 \).

Step 4: Check the other options.
Option A (92) and option C (91) do not match this count, they come from arithmetic slips, not a valid method. Since 90 is a clean, verified value, option D (none of these) is also wrong.

Final Answer:
The authorities need 90 different tickets. \[ \boxed{90} \]