College Algebra & Trigonometry, 2018a

306 CHAPTER 7. COMBINATORICS
different ways something can happen.
The first major idea of combinatorics is the fundamental principle of counting.
This is the idea that if two events occur in succession and there are m ways to do
the first one and n ways to do the second (after the first has occurred), then there
are m ∗ n ways to complete the two tasks in succession.
For example, in throwing two six-sided dice, there are 36 possibilities - six possibilities
from the first die and six from the second. These possibilities are listed
below:
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
If we were to throw a six-sided die and an eight sided-sided die, then there would
be 6 ∗ 8=48different possibilities.
Often the creation of a tree diagram can help us to visualize the possibilities.
Suppose that a three-digit code using the numbers 0, 1 and 2 is created so that
repetition of the numbers is allowed. There would 3 possibilities for the first
number, three for the second number and three for the third number meaning
that there would be a total of 3 ∗ 3 ∗ 3=27different possible codes.
The tree diagram to illustrate this is shown on the next page.