Fundamental Counting

How do you count?

Try counting these dots…

5

7

5×7= 35

So, there are 35 dots.

Is this any easier?

Problem 1 – What is for lunch?

How many different lunches can be made if a student must select one item from each category.

beverage main course dessert

Sprite hot dog apple

milk grilled cheese sandwich muffin

coffee bagel & cream cheese

egg salad sandwich

Problem 2 – PIN Numbers

Using only the digits 0 – 9, how many different 4 digit PIN numbers can be made?

Problem 3 – License Plates

Each license plate uses 4 letters followed by 3 numbers. How many different license plates can be made in this pattern?

The Fundamental Principle of Counting

If you have a number of decisions to make, each of which is not influenced by the results of the other decisions, the total number of ways is the product of the number of possibilities for each decision.

Example 1:

A B

This object is called a “directed graph” – it has nodes and arrows that connect them.

How many routes are therefrom A to B?

Example 1:

A B

How many routes are therefrom A to B?

1 choice 1 choice

3 choices

2 choices

From the Fundamental Principleof counting, you might guess1×3×1×2=6, but you would bewrong.

Example 1:

A B

How many routes are therefrom A to B?

1 choice 1 choice

3 choices

2 choices

AND AND

OR

There is a difference between“AND” choices and “OR” choices!

You can go by the 1×3×1 = 3 upper paths OR by the 2 lower paths. 3+2 = 5.

There are 5 paths.

Additive Counting Principle

If a task options that DO influence each other (i.e. choosing one item excludes choosing a different item),

the total number of choices is m+n+p+…

Mutually Exclusive Options

To add or multiply?

If it is an AND choice, you multiply,

if it is an OR choice you add.