Hi Sam,
Let's look at some smaller problems.
A fair fourfaced die with faces numbered 1,2,3 and 4 is tossed once and the number is recorded. How many numbers are possible?
This is almost too easy. The possible numbers are 1, 2, 3 and 4.
A fair fourfaced die with faces numbered 1,2,3 and 4 is tossed twice and the sequence of numbers is recorded. How many sequences are possible?
Each of the possible numbers in the previous problem could be followed by a 1, 2, 3 or 4. Thus the possible sequences are
11 
12 
13 
14 
21 
22 
23 
24 
31 
32 
33 
34 
41 
42 
43 
44 
Thus there are 4 4 = 4^{2} = 16 possible sequences.
A fair fourfaced die with faces numbered 1,2,3 and 4 is tossed three times and the sequence of numbers is recorded. How many sequences are possible?
Each of the possible numbers in the previous problem could be followed by a 1, 2, 3 or 4. Thus there are four times as many three number sequences as there are two number sequences. Thus there are 4^{2} 4 = 4^{3} three number sequences. A slightly different way to say this is that there are 4 options for the first number, 4 options for the second number and 4 options for the third number and hence 4 4 4 = 4^{3} possibilities in total.
Now try your problem.
Penny
