Question

There are 10 stations between A and B. A train is to stop at three of these 10 stations. The probability that no two of these stations are consecutive is

• A. $\frac{{}^8{C}_3}{{}^{10}{C}_3}$
• B. $\frac{{}^9{C}_3}{{}^{10}{C}_3}$
• C. $\frac{{}^7{C}_3}{{}^{10}{C}_3}$
• D. $\frac{{}^{10}{C}_3}{{}^{12}{C}_3}$

Answer 1

The correct option is A $\frac{{}^8{C}_3}{{}^{10}{C}_3}$

Select the three stations where the train will stop, say P, Q, R. There are now $4$ gaps among the stations (including the two ends) where the remaining $7$ stations may be distributed. Place $1$ station between P and Q and $1$ station between Q and R. There are remaining $5$ stations. These need to distributed among the $4$ gaps.

This can be done in: ${}^8{C}_3$ ways.

Total number of ways of selecting $3$ stations $=$ ${}^{10}{C}_3$
Hence, probability $=$ $\frac{{}^8{C}_3}{{}^{10}{C}_3}$