The easiest way,
Probabilty of getting 7 is 1/6 [(1,6),(2,5),(3,4)]*2=6/36=1/6
p(Sum=7)=1/6 and thus, p(Sum is not 7)=5/6
Now, imagine 3 places,
_ _ _
Case-1: We get one time sum=7 ans two time not,
_7 _ _
so, there can be 3 cases, 3*(1/6)*(5/6)*(5/6)
Case-2: We get two time sum=7 ans one time not,
_7 _7 _
so, there can be 3 cases, 3*(1/6)*(1/6)*(5/6)
Case-3
We get 7 at all positions,
_7 _7 _7
and this is only one case, 1*(1/6)*(1/6)*(1/6)
Now, these all are 'OR' cases so add them,
3*(1/6)(5/6)(5/6)+3*(1/6)(1/6)(5/6)+(1/6)(1/6)(1/6)
=75+15+1/216
= 91/216
P.S. This is just a binomial method only, but in GATE during the process of solving questions remembering formulas can be hard and we solve all the questions iteratively like I did here.