Question

Three numbers are chosen from $1$ to $30$. The probability that they are not consecutive, is

• A. $\frac{142}{145}$
• B. $\frac{144}{145}$
• C. $\frac{143}{145}$
• D. $\frac{1}{145}$

Answer 1

The correct option is B $\frac{144}{145}$

Probability that the numbers are not consecutive $=1-$probability of the numbers being consecutive.

Given numbers from $1to30$

$\therefore$ sample space is $30!$

Making groups of $3$ consecutive numbers from $(1,2,3)$ to $(28,29,30)$ we will get $28$ pairs.

Considering $3$ numbers as single digit, the numbers will be $28$ and those three numbers can be arranged in $3$ different ways.

Three numbers consecutively $=28!\times 3!$

$\therefore Probability=\frac{28!\times 3!}{30!}=\frac{1}{145}$

Probability that the numbers are not consecutive $=1-\frac{1}{145}=\frac{144}{145}$.