Out of 11 consecutive natural numbers
Question:

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:

1. (1) $\frac{15}{101}$

2. (2) $\frac{5}{101}$

3. (3) $\frac{5}{33}$

4. (4) $\frac{10}{99}$

Correct Option: , 3

Solution:

For an A.P. $2 b=a+c$ (even), so both $a$ and $c$ even

numbers or odd numbers from given numbers and $b$ number will be fixed automatically.

Required probability $=\frac{{ }^{6} C_{2}+{ }^{5} C_{2}}{{ }^{11} C_{3}}=\frac{25}{165}=\frac{5}{33}$