3 fair dice are rolled together 108 times. Then the expected number of times the 3 dice shows up different numbers, which can form A.P is__

 $n\left(S\right)=6^3$
Let  $p=P$  (getting 3 different numbers in A.P) in one trial
$(1,2,3),(2,3,4),(4,5,6),(1,3,5),(2,4,6)$  can occur on 3 dice (in any order)
Number of favorable case  $=6\times \overline{)3}=6^2$
 $p=\frac{1}{6}$
Let X=Number of times we get 3 different number in A.P out of 108 trials
Then  $X~B[n=108,p=\frac{1}{6}]$
 $E\left(x\right)=np=18$