15. Three dice, each showing numbers 1 to 6 are coloured red, blue and yellow respectively. Each of the dice is rolled once. The total of the numbers rolled is 10 . In how many different ways can this happen?
A 36
B 30
C 27
D 24
E 21
SolUtion
C
We note first that, because the dice are coloured, two outcomes with total 10 , but with different numbers rolled on particular dice, count as being different. For example
red : 6 blue : 3 yellow : 1
counts as being different from the outcome
red : 6 blue : 1 yellow : 3 .

With three different numbers there are three choices for the dice which rolls the first number, then two choices for the dice which rolls the second number, leaving just one choice for the dice which rolls the third number. This gives a total of $3 \times 2 \times 1=6$ arrangements for the three numbers.
It can be checked that when two of the numbers are the same these can occur in 3 different ways.
It is not possible to have three equal scores with total 10 .
To solve this problem we now list in the following table all possible ways a total of 10 may be obtained by throwing three dice. In each row of the table we also give the number of different ways the three numbers in the row may be arranged between the three dice.
Therefore the total number of different ways of achieving a total of 10 is $6+3+6+6+3+3=27$.