8. A mathematician rolls three six-sided dice (one red, one green, and one blue)

Question

8. A mathematician rolls three six-sided dice (one red, one green, and one blue). Or rather, she intends to roll them, but gets distracted by thoughts of combinatorics. (a) How many different possible rolls are there? (By a "roll", we mean the values rolled on the three dice. For example, one possible roll is "1 on red, 4 on blue, 4 on green".) (b) In how many of these rolls do all three dice have the same value? (c) In how many rolls do all three dice have different values? (d) In how many rolls do exactly two dice have the same value?

Explanation / Answer

There are three dices of colours red, blue and green;

a) total different possible rolls would be the total possible outcome or combinations;
Let the outcome on three dices by x, y, z (red , blue and green dice respectively)
x can be 1 to 6 = 6 values
y can be 1 to 6 = 6 values
z can be 1 to 6 = 6 values

Thus, total number of possible combinations = 6*6*6 = 216;
Thus, a total of 216 different possible rolls are there

b) A dice can have numbers only from 1 to 6;
Thus, for all the three dices to have the same outcome, they all can have all 1s, all2s all 3s, all 4s, all 5s, all 6s
Thus, a total of 6 rolls are possible in which all three dice have the same value;

c) for all three to have different value
x,y,z
x can take 6 values;
y can take all 6 except x's value so y can take 5 values
z can take all 6 except x and y 's values (which are distinct) so z can take 4 values
so possible number of rolls would be x*y*z = 6*5*4 = 120 values

d) when 3 dice are rolled either
i ) all three are same = 6 rolls
ii) any 2 dices are same = 'm' rolls
iii) all 3 are different = 120 rolls

Hence 216 = 6+m+120
m = 216-126 = 90 rolls

Hence in 90 rolls exactly two dices have the same value

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