Question

There are $15$ stations on a certain railway line. How many different types of tickets must be printed so that a passenger can travel from one station to another in that line $?$

Answer 1

Apply the combination formula:

The combination is the number of ways of selecting objects such that the order of selection is not to be counted.

${}_n{C}_r=\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{{}_n{P}_r}{r!}=\frac{n!}{r!(n-r)!}$

From a single station if he travels to $14$ stations he needs $14$ different tickets.

The number of ways of choosing $2$ station from $15$ different stations:

No of ways

$$\begin{gathered} ={}^{15}\mathrm{C}_2 \\ =\frac{15\times 14}{2} \\ =105 \end{gathered}$$

The number of different types of tickets :

$$\begin{gathered} =2\times 105 \\ =210 \end{gathered}$$

Hence, $210$ types of tickets must be printed.