Suppose we roll three tetrahedral dice that have 1 2 3 and 4

Suppose we roll three tetrahedral dice that have 1, 2, 3, and 4 on their four sides. Find the distribution for the sum of the three numbers.

Distriubtion of sum of 3 tetrahedral dice:

P(Sum = 3) = (1/4)*(1/4)*(1/4) = 1/64 [All 1s in the 3 dices]

P(Sum = 4) = (1/4)*(1/4)*(1/4) * (3) = 3/64 [2 in 1 dice and 1 in the other 2]

P(Sum = 5) = (1/4)*(1/4)*(1/4) * (3 + 3) = 6/64 [3 in 1 dice and 1 in other 2 dices, and 2 in 2 dices and 1 in the 3rd dice]

Similarly,

P(Sum = 6) = (3+6+1)/64 = 10/64 [ 4 in 1 dice and 1 in the other 2, and 1,2,3 in 3 dices, and 2 in all 3 dices]

P(Sum = 7) = (6 + 3 + 3) / 64 = 12/64 [ 1,2,4 in 3 dices, and 2 in 2 dices and 3 in the 3rd dice, 3 in 3 dices and 1 in the 3rd dice]

P(Sum = 8) = (3 + 6 + 3)/64 = 12/64 [ 2 in 2 dices and 4 in the 3rd dice, and 3 in 2 dices and 2 in the 3rd, and 1,3,4 in 3 dices]

P(Sum = 9) = (6 + 3 + 1)/64 = 10/64 [4,3,2 in 3 dices, and 4 in 2 dices and 1 in the 3rd, and 3 in all 3 dices]

P(Sum = 10) = (3 + 3)/64 = 6/64 = [4 in 2 dices and 2 in the 3rd and 3 in 2 dices and 4 in the 3rd]

P(Sum = 11) = (3)/64 [4 in 2 dices and 3 in the 3rd]

P(Sum = 12) = 1/64 [ 4 in 3 dices]