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.
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.
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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.
Updated On: Jul 13, 2026
- 49
- 42
- 21
- 7
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The Correct Option is B
Solution and Explanation
Step 1: Count the total number of stations.
The train runs between two end stations, Tatanagar and Howrah, and stops at five stations in between. So the total number of stations the train touches is
\[ 2 + 5 = 7 \]
Step 2: Understand what a different kind of ticket means.
A one-way ticket is defined by an ordered pair of stations: a starting station and a destination station. A ticket from station P to station Q is a different, separate ticket from one going from Q to P, since the journey direction is different. So we need to count ordered pairs of distinct stations out of the 7 stations.
Step 3: Apply the permutation count.
The number of ways to choose an ordered pair (origin, destination) from 7 distinct stations, with origin and destination different, is given by
\[ {}^{7}P_{2} = 7 \times 6 = 42 \]
This is because there are 7 choices for the origin station, and once that is fixed, 6 remaining choices for the destination station.
Step 4: Sanity check with a smaller case.
If there were only 2 stations (no stoppages), the formula would give \(2\times 1=2\) tickets: one for each direction, which matches common sense. This confirms the logic scales correctly to 7 stations.
Final Answer:
Indian Railways needs to print 42 different kinds of tickets. \[ \boxed{42} \]
The train runs between two end stations, Tatanagar and Howrah, and stops at five stations in between. So the total number of stations the train touches is
\[ 2 + 5 = 7 \]
Step 2: Understand what a different kind of ticket means.
A one-way ticket is defined by an ordered pair of stations: a starting station and a destination station. A ticket from station P to station Q is a different, separate ticket from one going from Q to P, since the journey direction is different. So we need to count ordered pairs of distinct stations out of the 7 stations.
Step 3: Apply the permutation count.
The number of ways to choose an ordered pair (origin, destination) from 7 distinct stations, with origin and destination different, is given by
\[ {}^{7}P_{2} = 7 \times 6 = 42 \]
This is because there are 7 choices for the origin station, and once that is fixed, 6 remaining choices for the destination station.
Step 4: Sanity check with a smaller case.
If there were only 2 stations (no stoppages), the formula would give \(2\times 1=2\) tickets: one for each direction, which matches common sense. This confirms the logic scales correctly to 7 stations.
Final Answer:
Indian Railways needs to print 42 different kinds of tickets. \[ \boxed{42} \]
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