Question

A train is going from Cambridge to London stops at nine intermediate stations. Six persons 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

For $S_1$, $9$ different tickets available, one for each of the remaining $9$ stations;
similarly at $S_2$, $8$ different tickets are available; so on.
Hence, it is clear, that the total no. of different tickets
$=9+8+7+6+5+4+3+2+1=45$
Hence the six different tickets must be any six of these $45$; there are
evidently as many different sets of $6$ tickets as there are combinations of $45$ things taken $6$ at a time.
Hence the required number $={}^{45}{C}_6=8145060$