Question

A train is going from Delhi to Indore, stops at nine intermediate stations. Six persons enter the train during the journey with six different tickets. The number of different sets of tickets possessed by them is

• A. ${}^{50}{C}_6$
• B. ${}^{54}{C}_6$
• C. ${}^{45}{C}_6$
• D. None of these

Answer 1

The correct option is C ${}^{45}{C}_6$

Let $D$ denotes Delhi and $I$ denotes Indore. There are $9$ intermediate stations. Let they be denoted by $s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9$

For $s_1$, $9$ different tickets are available, one for each of the $9$ remaining stations. Similarly for $s_2$ $8$ different tickets are available and so on.

Thus total number of $9$ different tickets $=9+8+7+6+5+4+3+2+1=45$

Thus the $6$ different tickets must be any $6$ of these $45$ and there are clearly as many different sets of $6$ tickets as there are combinations of $45$ things taken $6$ at a time.

So, required number is ${}^{45}{C}_6$.