Question

Steel Express runs between Tatanagar and Howrah and has five stoppages in between. Find the number of different kinds of one-way second class tickets that Indian Railways will have to print to service all types of passengers who might travel by Steel Express.

• A. 49
• B. 42
• C. 21
• D. 7

Hint:

Count the total stations (2 end points plus 5 stops), then count ordered pairs of stations since a ticket from P to Q differs from one from Q to P.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept.
A train ticket names one station as the start and a different station as the end. Two stations can be paired for travel in two distinct ways depending on which one is the origin, so this is really a two-part counting question: first count how many station-pairs exist, then account for direction.

Step 2: Key Formula or Approach.
Total stations $= 2$ terminal stations $+ 5$ stoppages $= 7$ stations.
The number of ways to pick any 2 different stations out of 7, without worrying about order, is given by the combination formula:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

Step 3: Detailed Explanation.
Each unordered pair of stations, say $\{P, Q\}$, actually needs two separate tickets printed: one for travel from $P$ to $Q$, and a separate one for travel from $Q$ to $P$, since these are different journeys with different origin and destination printed on the ticket. So the true number of distinct tickets is double the number of unordered pairs:
\[ 21 \times 2 = 42 \]

Step 4: Final Answer.
Counting unordered station-pairs and then doubling for direction gives the same result as directly counting ordered pairs: 42 different kinds of tickets are needed. \[ \boxed{42} \]