15.2 Multiplication, Addition and Complement Principles

The Multiplication Principle

• If an action can be performed in n1 ways, and for each of these ways another action can be performed in n2 ways, then the two actions can be performed in n1 n2 ways.

Actions being performed TOGETHER

MULTIPLY

• You have 2 shirts, 4 pairs of pants and 3 pairs of shoes, how many different shirt, pants and shoes outfits can be made?

The Addition Principle

• Two events that cannot be performed together are called MUTUALLY EXCLUSIVE.

• If two actions are mutually exclusive and the first can be done in n1 ways and the second in n2 ways, then one action OR the other can be done in n1+ n2 ways.

You have 2 blazers and 4 sweaters.A) How many blazer and sweater outfits can you wear?

B) Suppose you decide to wear either a blazer or a sweater, but not both. How many choices do you have?

Actions NOT being performed together

ADD

• In how many ways can 7 people line up?

Factorials

n! is defined as:

n(n-1)(n-2) … 3 2 1∙ ∙ ∙ ∙

• How many different 9 person lineups can you create with a baseball team of 11 players?

• How many license plates can be made using 3 letters followed by 3 digits?

• How many different license plates can be made using 6 letters and digits in any order?

• In Morse Code, the letters of the alphabet are represented by sequences of dots and dashes. Show that no more than 4 symbols (of dots and dashes) are needed to represent all of the alphabet.

The Complement Principle

If A is a subset of a universal set U, then

( ) ( ) ( )n A n U n A

The number of elements in set A = the number of elements in the universal set U – the number of elements NOT in set A

• Find the number of 4-digit numbers containing at least one digit 5. (The first digit cannot be zero)

• Given the digits 0,1,2,3,4,5 how many 4 digit numbers can be formed if (1st digit cannot be zero):

a. Digits can be repeated?

b. Digits cannot be repeated?

• How many different ways can you make a license plate with 3 letters followed by 2 digits with at least 1 Q?

Homework

• P. 575 (Written Exercises) 1 - 8