Be very careful

If you roll $x$ the first time there are $x-1$ valid options on each of the next two rolls and for both of the next two rolls to be valid there are $(x-1)^2$ combinations. There are $6^3=216$ total options hence the answer is

$\frac{1}{216}\sum\limits_{n=1}^{6} (n-1)^2=\frac{1}{216}\sum\limits_{n=0}^{5} n^2 = \frac{1}{216}\frac{5(5+1)(2(5)+1)}{6}=\boxed{\frac{55}{216}}$

Probability= number of favourable outcomes ÷ total number of outcomes.

Total number of outcomes= 6×6×6= 216

Favourable outcomes include-

(2,1,1)

(3,1,1) (3,2,2) (3,1,2) (3,2,1)

(4,1,1) (4,2,2) (4,3,3) + 2(3C2)

(5,1,1) (5,2,2) (5,3,3) (5,4,4) + 2(4C2)

(6,1,1) (6,2,2) (6,3,3) (6,4,4) (6,5,5) + 2(5C2)

Now counting the number of favourable outcomes gives 17 + 2(3C2 + 4C2 + 5C2)

= 17 + 2(3+6+10) = 17+ 38 =55

Therefore the probability of this event is 55/216

55/216....as if 1st throw can be minimum 2 and second and third dice roll would be compulsory 1 (1/6 1/6 1/6)...similary if 1 st throw is 3 then second and third dice can be 1 or 2 (1/6 2/6 2/6)...similarly for if first throw is 4 then(1/6 3/6 3/6) as 2nd and 3 rd throw can take calue 1,2,3...so on till 1 st throw be 6... 1/6 1/6 1/6+1/6 2/6 2/6+1/6 3/6 3/6+1/6 4/6 4/6+1/6 5/6 5/6=55/216.

I've encountered this kind of problem before and I just use the property of the sequence it shows : $1^{2}, 2^{2}, ... 5^{2}$ . Thus, it has a pattern of sum of squares which is $n \times (n+1) \times (2n+1) / 6$ and divide it with the total space = 216 final answer would be 55/216

Let's take a look at all possible first rolls and consequent probability of getting a higher number.

If first roll is a 1,

Probability of getting a higher number in any one of the next rolls (POH) = 5/6.

Probability of getting a higher number in the next two rolls combined (POH^2) = 5/6 * 5/6 = 25/36.

If first roll is a 2,

POH = 4/6

POH^2 = 16/36.

For 3,4,5,6 we get 9/36, 2/36/, 1/36 and 0/36 as POH^2 respectively.

Adding all probabilities and taking average we get.

$\frac { 25+16+9+4+1+0 }{ 36*6 } = \frac { 55 }{ 216 }$