Question

Three numbers are chosen from $ 1 $ to $ 30 $ . Find the probability that they are not consecutive.

• A. $ \frac{16}{81} $
• B. $ \frac{144}{145} $
• C. $ \frac{80}{145} $
• D. $ \frac{65}{81} $

Answer 1

Answer: (B)

The total number of ways in which $ 3 $ numbers can be chosen out of $ 30 $ numbers $ {{=}^{30}}{{C}_{3}}=4060 $ . The number of ways of choosing $ 3 $ consecutive numbers is $ 28 $ .
Therefore, the number of ways in which the three numbers chosen are not consecutive is $ 4060-28=4032 $ .
Hence, the required probability $ =\frac{4032}{4060}=\frac{144}{145} $