There are 15 train stops between Chennai and Assam. How many train tickets are to be printed, so that a person can travel between any of the two stations (irrespective of direction of travel)?

Key Concepts Needed

1. Total Stations
   • Intermediate stops = 15
   • Terminals = Chennai + Assam = 2
   • Hence, $n = 15 + 2 = 17$ stations.

2. Unordered Pairs (Combinations)
   A ticket simply specifies a start and an end station; direction doesn’t matter. Thus we want the count of unordered pairs, written mathematically as "17 choose 2" or $\binom{17}{2}$.

3. Combination Formula
   $\binom{n}{2} = \frac{n(n-1)}{2}$
   This counts how many ways we can pick 2 distinct stations out of $n$.

Logical Chain to Solve

1. Find $n$. Identify the total number of stations (terminals + intermediate).
2. Decide on order or no order. Since a ticket A→B and B→A are identical in wording, the order does not matter.
3. Apply combination formula. Use $\binom{n}{2}$ to calculate the distinct pairs.
4. Compute. Plug in $n=17$ and simplify.

🤔 What’s the problem?

Imagine you have many railway stations on a line. You need one unique ticket for every pair of stations, because a traveller may start at any station and get off at any other.

🏃‍♂️ How to think like a 10-year-old?

1. Count the stations first.
   We’re told there are 15 stops in-between Chennai and Assam. Add the two end stations:
   15 (middle) + 2 (ends) = 17 stations in total.
2. Pair them up.
   To make sure any two stations can be connected, list every possible pair of stations.
3. No double-counting!
   A ticket from Station A to Station B is the same as from B to A, so we only count each pair once.
4. Magic counting shortcut.
   There’s a quick formula for the number of pairs in a group of $n$ things:
   $\text{Pairs} = \frac{n(n-1)}{2}$
5. Plug $n = 17$:
   $\frac{17\times16}{2} = 136$

So, 136 different tickets cover every possible trip!

Step-by-Step Solution

1. Total number of stations
   $n = 15 + 2 = 17$

2. Formula for number of unordered pairs
   $\binom{n}{2} = \frac{n(n-1)}{2}$

3. Substitute $n = 17$

   $\binom{17}{2} = \frac{17 \times 16}{2}$

4. Simplify

   $\frac{17 \times 16}{2} = 17 \times 8 = 136$

5. Answer

   136 tickets are required so that any two stations can be connected in either direction.