Here is a new method( ans is 7/72)

 

if a dice is rolled 4 times such that each num is no smaller than the preceding num=each num is greater than equal to preceding num

if a case is (x1,x2,x3,x4), x1<=x2<=x3<=x4

num of ways of taking group of 4 from 1-6 is 6c4=15

Hence in each of these combination case there is only one arrangement such that they are in ascending order.this forms a one one correspondence.

So the num of ways of taking groups of 4 such that the elements are in ascending order is the same 15.

A = can be chosen in 3c1=3 ways

suppose x1=x2<x3<x4

no of ways of taking groups of 3 from 1-6 is 6c3=20

in each of these cases there is only one arrangement such that they are in ascending order.In one such ascending order case if 1st element is repeated it forms num of ways of aranging x1=x2<x3<x4 would be the same 20.

total num of ways is 20*3=60

 two = can be chosen in 3c2=3 ways . So num of ways= 3*6c2=45

num of ways is 6

so in all total =15+60+45+6=126

probablity is 126/6^4=7/72