ORGANIZED COUNTING
(The Fundamental Principle of Counting)
COMBINATORICS: the systematic ways of counting arrangements or
outcomes, especially in complex situations.
TREE DIAGRAMS
 each branch represents a different choice
 useful for organizing small amounts of information only
Example 
You are hungry!! Create a sandwich for lunch using
sliced bread or a kaiser; ham, turkey, or roast beef;
and mustard or mayonnaise.
How many different sandwiches are possible?
BOX/SLOT METHOD
 each box/slot represents a different action
 the number inside the box/slot represents the number of
choices for that action

3 actions or tasks 
bread
meat condiment
FUNDAMENTAL COUNTING PRINCIPLE
(or MULTIPLICATION PRINCIPLE)
If an action can be done in m ways and for each way a second
action can be done in n ways, then the two actions can be
performed, in that order, in mn ways.
Example 
A store sells 6 different computers, 4 different
monitors, 5 different printers, and 3 different
multimedia packages. Determine the number of
different computer systems that are available.
Example 
A coin is flipped six times. Determine the number of
possible outcomes.
In some situations, an indirect method makes a calculation easier.
Example 
Method 1: Direct
Method 2: Indirect
Tom, the triathlete, has four pairs of running shoes
loose in his gym bag. Determine the number of ways
he can pull out two unmatched shoes one after the
other.
In some situations, subsets of possibilities must be counted separately.
Example 
Sailing ships used to send messages with signal flags
flown from their masts. Determine the number of
different signals that are possible with a set of four
distinct flags if a minimum of two flags is used for each
signal.
Mutually exclusive actions are those actions that could not occur at the
same time. The additive counting principle is applied in these cases.
ADDITIVE COUNTING PRINCIPLE
(or RULE OF SUM)
If one mutually exclusive action can occur in m ways, a second in n
ways, a third in p ways, and so on, then there are m + n + p + …
ways in which one of these actions can occur.
Example 
Determine the number of ways a sum of 3 or a
sum of 10 can be rolled with a pair of dice.
HOMEWORK:
p.229–230 #1–12
(Correct answer to #7 is 24)