Subject: What is the easiest way to solve this problem?
Name: Sam
Who are you: Student

A fair eight-faced die with faces numbered 1,2,3,4,5,6,7 and 8 is tossed six times and the sequence of numbers is recorded. How many sequences are possible?

What is the easiest way to solve this problem?

Hi Sam,

Let's look at some smaller problems.

A fair four-faced 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 four-faced 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 = 42 = 16 possible sequences.

A fair four-faced 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 42 4 = 43 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 = 43 possibilities in total.