Total number of balls in box $=6$ red balls $+ 7$ green balls $+ 5$ blue balls $=18$ balls
Probability of selecting red ball $=\frac{6}{18}$
The probability of selecting the smallest red ball (it is given that each ball is of different size) $=\frac{6}{18} \times \frac{1}{6}$
So probability that the red ball selected is the smallest red ball $= \dfrac{\text{Probability of red ball being selected AND the selected red ball being the smallest}}{\text{Probability of red ball being selected}}$
$\qquad \qquad = \dfrac{\frac{6}{18}\times \frac{1}{6}}{\frac{6}{18}}=\frac{1}{6}$
Option $(C)$ is correct.