Question

A train is going from Chennai to Bangalore and make $6$ intermediate stops on the way. Three persons enter the train after it has started from Chennai each with $3$ different tickets. How many different sets of tickets may they have had?

• A. ${}^{21}{P}_3$
• B. $3\times {}^{21}{C}_3$
• C. ${}^{21}{C}_3$
• D. $3\times {}^{15}{C}_3$

Answer 1

The correct option is C ${}^{21}{C}_3$

Since, the three persons enter the train after it has started from Chennai, so they could have entered at:

$1\text{st}$ station (from where they could have bought tickets for the $2\text{nd},$ $3\text{rd},$ $4\text{th},$ $5\text{th}$ or $6\text{th}$ station or for Bangalore i.e. $6$ tickets).

$2\text{nd}$ station (from where they could have bought tickets for the $3\text{rd},$ $4\text{th},$ $5\text{th}$ or $6\text{th}$ station or for Bangalore i.e. $5$ tickets).

$3\text{rd}$ station (from where they could have bought tickets for $4\text{th},$ $5\text{th}$ or $6\text{th}$ station or for Bangalore i.e. $4$ tickets).

$4\text{th}$ station (from where they could have bought tickets for the $5\text{th}$ or $6\text{th}$ station or for Bangalore i.e. $3$ tickets).

$5\text{th}$ station (from where they could have bought tickets for the $6\text{th}$ station or for Bangalore i.e. $2$ tickets).

$6\text{th}$ station (from where they could have bought a ticket for Bangalore i.e. $1$ tickets).

Thus, we can see that there are $6+5+4+3+2+1=21$ tickets available out of which $3$ tickets are to be selected.
This can be done in ${}^{21}{C}_3$ ways.