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Q.
There are $12$ railway stations in a railway line. A train stops at $4$ stations randomly. Then probability that exactly three of them are consecutive is
Probability
Solution:
Let A be the event that out of $4$ stations $3$ are consecutive.
$\therefore n\left(A\right) = 2 \times 8 + 8 \times 7$
$= 16 + 56$
$= 72$
[Out of $4$ stations there are $2$ ways in which three consecutive stations occurs]
$n\left(S\right)$ = Ways of selected $4$ stations out of $12 = \,{}^{12}C_{4}$
$\therefore P\left(A\right) = \frac{72 \times4 \cdot3 \cdot2 \cdot1}{12 \times11 \times10 \times9} = \frac{8}{55}$
