Section 4.3
Trees and Counting Techniques

I. Tree Diagrams

When calculating probabilities, you need to know the total number of ______in the ______.

Example 1: Using a Tree Diagram

Use a tree diagram to determine the sample space of 2 coin flips.

Tree

Example 2: Using a Tree Diagram

Terrence wakes up and opens his closet. Inside are two pairs of shoes (one black and one brown), two pairs of pants (jeans and khakis), and three shirts (t-shirt, polo shirt, and button down). List all possible outfits Terrence can wear.

Tree

II. Multiplication Rule of Counting

The size of the sample space is the ______ of our probability.

So we don’t always need to know what each outcome is, just the ______ of outcomes.

Multiplication Rule of Compound Events:

If…

X = total number of outcomes for event A

Y = total number of outcomes for event B

Then number of outcomes for A followed by B =

Example: Terrence’s outfits

(# of shoes) * (# of pants) * (# of shirts) = # of outfits.

III. Permutations

Sometimes, we are concerned with how many ways a group of objects ______:

  • How many ways
  • How many ways
  • How many ways

Example:

Miss Roberts needs to sit five students in five chairs. How many ways can she arrange them?

Think of each chair as an EVENT:

Event / Possibilities
Chair 1
Chair 2
Chair 3
Chair 4
Chair 5

Use the Multiplication Rule:

FACTORIAL

+ denoted with ! 

+ Multiply all integers ≤ the number  5! =

+ 0! =

+ 1! =

+ Calculate 6! =

+ What is 6! / 5!?

A ______ gives all of the possible ______of n objects taken r at a time.

Permutation Formula:

You have ______

You select______

This is the number of ways you could select and arrange in order:

Another common notation for a permutation is nPr.

Example.

If Miss Roberts had 9 students but only five chairs, how many ways could she seat them?

n = ____r = _____

=

There are ______ ways to arrange 9 students in 5 chairs.

IV. Combinations

Sometimes, we are only concerned with ______a group and ______ in which they are selected.

A combination gives the number of ways to ______of r objects from a group of size n.

Combination Formula

You have ______

You want a group of ______

You ______what order they are selected in

Combinations are also denoted nCr

Example.

Mrs. Roberts has a class of 28 kids. She needs 3 students to clean her boards after school. How many groups of 3 could she get out of the 28 students?

=

She could make ______different groups of 3 students.

When to use what:

ORDER MATTERS 

ORDER DOESN’T MATTER 

Created By: J& J Productions