As each dice has six sides, the probability of getting any one side or any one number is 1/6. Getting red or blue or green does not in any way relies on the other, but independent. So we should multiply the independent probabilities.
a) The probability of getting 6(R), 6(G), and 6(B).
The probability of getting 6(R) is 1/6 and 6(G) is 1/6 and 6(B) is 1/6. So,
(1/6)(1/6)(1/6) = (1/6)3
= 1/216
b) The probability of getting the outcome 6(R), 5(G), and 6(B).
The probability of getting 6(R) is 1/6 and 5(G) is 1/6, and 6(B) is 1/6
(1/6)(1/6)(1/6) = 1/216
c) The probability of getting the outcome 6(R), 5(G), and 4(B)
The probability of getting 6(R) is 1/6, 5(G) is 1/6, and 4(B) is 1/6
(1/6)(1/6)(1/6) = 1/216
d) No sixes at all
This is the same as getting anything but sixes.
That is (5/6)(5/6)(5/6) = 125/216
e) The probability of getting different numbers in all the dice.
For this we have to consider each die separately. The first dice thrown can be any number. So the probability is 1.
The second dice thrown can be any number other than the number in the first dice. So the probability of not getting the same number is 1- 1/6 = 5/6.
The third dice thrown can be any number except the two numbers got in the previous both the dice. So the probability is as follows:
1- (1/6 + 1/6) = 1- 2/6
= 2/3.
Therefore, the probability of all different dice is as follows:
(1)(5/6)(2/3) = 10/18
= 5/9.