For this we have to find number of ways we can get a sum of 10 in 3 rolls of dice which is nothing but the number of positive integral solutions of the equation :
x1 + x2 + x3 = 10 where 1 <= x1,x2,x3 <= 6
Now number of possible integral solutions can be found as :
Coefficient of x10 in (x + x2 + ........ + x6)3
==> Coefficient of x10 in x3 (1 + x +.....+ x5)3
==> Coefficient of x7 in (1 + x +.....+ x5)3
==> Coefficient of x7 in (1 - x6)3 . (1 - x)-3
Now only relevant terms will be extracted from (1 - x6)3 and accordingly powers of (1 - x)-3 will be taken to get the coefficients of x7 .
So (1 - x6)3 = (1 - 3x6 + 3x12 - x18) out of which only first 2 terms are relevant..
So , due to the term "1" , coefficient of x7 is needed in (1 - x)-3 which is : 3-1+7C7 = 9C7 = 36
due to term "x6 " , coefficient of x is needed in (1 - x)-3 which is : 3-1+1C1 = 3C1 = 3
So due to -3x6 , we will have coefficient = -3 * 3 = -9
Hence net coefficient of x7 = 36 - 9 = 27
Hence P(sum of 3 dices is 10) = n(sum of 3 dices is10) / Total outcome
= 27 / (6 * 6 * 6)
= 27 / 216
= 1 / 8