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. $\frac{{{}^{10}C}_3}{{{}^{10}C}_3}$

Answer 1

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

$xS1$ $yS2$ $zS3w$

$\hookrightarrow$ slops $\hookleftarrow$

No. of stations $=x+1+y+1+z+1+w=10$

$\Rightarrow x+y+z+w=7$

$x,w\ge 0,y+z\ge 1$

$\Rightarrow x+y+z+w=5$

$x,y,z,w\ge 0$

$\therefore$ Number of ways $=\frac{\angle 5+(4-1)}{\angle 5\angle 4-1}$

$=\frac{\angle 8}{\angle 5\angle 3}={}^8{C}_3$

Probability $=\frac{{}^8{C}_3}{{}^{10}{C}_3}$