Step 1:

Let $X$ be the discrete RV denoting the number of sixes when three dice are thrown once.

$X$ takes the values $0,1,2,3$

Step 2:

$P(X=0)$=probability that all three dice do not show 6

$\qquad\;\;\;\;\;=\large\frac{5}{6}\times\frac{5}{6}\times\frac{5}{6}=\frac{125}{216}$

$P(X=1)$=probability of one dice showing 6

$\qquad\;\;\;\;\;=3C_1\large\frac{1}{6}\times\frac{5}{6}\times\frac{5}{6}=\frac{75}{216}=\frac{25}{72}$

$P(X=2)$=probability of two dice showing 6

$\qquad\;\;\;\;\;=3C_2\large\frac{1}{6}\times\frac{1}{6}\times\frac{5}{6}=\frac{15}{216}=\frac{5}{72}$

$P(X=3)$=probability of all three dice showing 6

$\qquad\;\;\;\;\;=\large\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}=\frac{1}{216}$

Step 3:

The probability distribution is given by