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# Three numbers are chosen from 1 to 20.Find the probability that they are not consecutive.

$\begin{array}{1 1}(A)\;\large\frac{186}{190}\\(B)\;\large\frac{187}{190}\\(C)\;\large\frac{188}{190}\\(D)\;\large\frac{18}{20C_3}\end{array}$

Step 1:
Probability that the numbers are not consecutive =1-probability of the numbers being consecutive
Given numbers from 1 to 20
$\therefore$ Sample space =20!
3 numbers consecutively =$18!\times 3!$
Considering 3 numbers as a single digit.
$\therefore$ The numbers will be 18 and those three numbers can be arranged in 3! ways
$\therefore$ Probability =$\large\frac{18!\times 3!}{20!}$
$\Rightarrow \large\frac{3}{190}$
Step 2:
Probability that they are not consecutive
$\Rightarrow$ 1-Probability that they are consecutive
$\Rightarrow 1-\large\frac{3}{190}=\frac{187}{190}$
Hence (B) is the correct answer.