Question

A train is going from Cambridge to London stops at nine intermediate stations. Six people enter the train during the journey with six different tickets. Let k be the number of different sets of tickets they have had. Find the difference of largest and smallest digit of k?

Answer 1

Hint: In this question, we first need to find the possibilities a passenger can have to get down when he boards at a particular station. Then the sum of all these possibilities gives the total number of ways a passenger can have a ticket. Now, we need to find the total number of combinations possible using the formula for number of combinations possible which is given by \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\] for six passengers and simplify it further. Then the difference between the largest and smallest digit in the result gives the value of k.

Complete step-by-step answer:
Let us name the intermediate stations between Cambridge and London as
S1, S2 , S3,, S4, S5, S6 , S7, S8, S9
Now, let us consider a passenger boards the train at Cambridge
Number of ways for passenger to get down are 10
Now, let us consider the passenger boards at S1
Then the number of ways to get down are 9
Now, let us consider the passenger boards at S2
Then the number of ways for the passenger to get down are 8
Now, let us consider the passenger boards at S3
Then the number of ways for him to get down are 7
Now, let us consider the passenger boards at S4
Then the number of ways to get down are 6
Now, let us consider the passenger boards at S5
Then the number of ways to get down are 5
Now, let us consider the passenger boards at S6
Then the number of ways the passenger have to get down are 4
Now, let us consider the passenger boards at S7
Then the number of ways for the passenger to get down are 3
Now, let us consider the passenger boards at S8
Then the number of ways for him to get down are 2
Now, let us consider the passenger boards at S9
Then the number of ways the passenger can get down are 1
Now, the total number of ways possible to get a ticket are
\[\Rightarrow 10+9+8+7+6+5+4+3+2+1\]
Now, on further simplification we get,
\[\Rightarrow 55\]
Thus, there are 55 ways possible to get a ticket by the passenger.
Now, the total number of sets of ticket can be bought by 6 passengers is given by
The number of combinations possible by six passengers to get ticket
\[\Rightarrow {}^{55}{{C}_{6}}\]
As given in the question that it is represented by k we get,
\[\Rightarrow k={}^{55}{{C}_{6}}\]
As we already know that
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
Now, on simplifying k using the above formula we get,
\[\Rightarrow k=\dfrac{55!}{49!6!}\]
Now, on simplifying it further we get,
\[\therefore k=2,89,89,675\]
Thus, the largest digit in k is 9 and smallest digit in k is 2
The difference between these two is given by
\[\begin{align}
  & \Rightarrow 9-2 \\
 & \Rightarrow 7 \\
\end{align}\]
Hence, the difference between largest digit and smallest digit of k is 7.

Note:It is important to note that there are different cases possible for the passenger to board and get down the train which changes according to the station he boards. So, we need to consider all those cases and add them to get the total cases possible.It is also to be noted that we need to calculate the number of combinations possible as there are 6 passengers and 9 intermediate stations. We should not neglect any of the case because it changes the result.