First slide

Combinations

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

Moderate

A

$^{50} C_{6}$
B

$^{54} C_{6}$
C

$^{45} C_{6}$
D

None of these

Solution

Let D denotes Delhi and I denotes Indore.
For S₁, 9 different tickets are available, one for each of the remaining 9 stations. Similarly, for S₂, 8 different tickets are available and so on.
Therefore, total number of 9 different tickets = 9 + 8 + 7 + 6 + 5 + 4 +3 +2 +1 = 45
Thus, the six different tickets must be any six of these 45 and there are clearly as many different sets of 6 tickets as they are combinations of 45 things taken 6 at a time.
Hence, the required number $=^{45} C_{6}$ .

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