Dice probabilities

[Albert Chan]

Instead of multiply to expand polynomial, we could also do division.

(x^2+x+1)^5 = (x^3-1)^5 / (x-1)^5

Numerator coefficients is simply 5th row of pascal triangle, with alternate sign:

x^15 - 5x^12 + 10x^9 - 10x^6 + 5x^3 - 1

Synthetic Division, (x-1) 5 times, we get the expanded polynomial needed.
With symmetry, we only need 2 terms. (for exponent >= 7, we don't even need last 2 columns)

Code:

1> 1   0   0   -5    0    0 
1> 1   1   1   -4   -4   -4
1> 1   2   3   -1   -5   -9
1> 1   3   6    5    0   -9
1> 1   4   10  15   15    6 
   1   5   15  30   45   51

P(sum >= 7) = (1+5+15+30)/3^5 = 17/81 ≈ 0.2099

Or, we skip synthetic divisions, and go straight for answer

P(sum >= 7) = (1+5+T₅+(Te₅-5))/3^5 = (1+T₅+Te₅)/3^5 = (1+comb(6,2)+comb(7,3))/3^5 = 17/81

https://en.wikipedia.org/wiki/Triangular_number
https://en.wikipedia.org/wiki/Tetrahedral_number