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Warm Up. How do you think you did on your test? How did you prepare for your test? What are some things you can do differently next time?. The Fundamental Counting Principle. You’re planning a date: dinner, entertainment, and dessert.

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Warm up
Warm Up

How do you think you did on your test?

How did you prepare for your test?

What are some things you can do differently next time?



You re planning a date dinner entertainment and dessert
You’re planning a date: dinner, entertainment, and dessert.

  • You have two choices for dinner: Happy Meals at McDonald's or microwave burritos from the local Quickymart. 

  • You have three choices for entertainment:  bowling, a movie or watching wrestling on TV. 

  • You have two choices for dessert:  s'mores and pie.  How many possible dates are there?

When in doubt, make a chart!


Do you really want to make a chart every time
Do you really want to make a chart every time?

__________________

Event 1: Dinner

__________________

Event 2: Entertainment

__________________

Event 3: Dessert


Fundamental counting principle
Fundamental Counting Principle

Keep doing this for all the events that occur

# of ways the second event can occur

# of ways the first event can occur

# of ways the third event can occur


Example 1
Example 1

  • A fast food restaurant sells: hot dogs, hamburgers, chicken sandwiches, and barbecue sandwiches. They offer as sides: French fries, hushpuppies, or onion rings.

  • How many possible combinations are there?

Multiply those bad boys.

__________________

Event 2: Choose a side.

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Event 1: Choose an entrée.


Example 2
Example 2

  • A mechanic offers three types oil changes: standard, synthetic, and high mileage; two types of wiper blades: low profile, and heavy use; and two types of mufflers: chrome and matte black.

  • How many possible combinations are there?


Example 3
Example 3

An ice cream store offers three types of cones and 31 flavors. How many different single-cone ice-cream cones is it possible to buy at this store?


Example 4 a little more challenging
Example 4: A little more challenging!

In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if letter repetition is allowed?


Example 5 stepping it up
Example 5: Stepping it up!

In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letters is not allowed?


Factorial notation
Factorial Notation!

  • Denoted by n! and is called “n factorial”

  • Only works if the number of items is equal to the number of spots being filled

    3! = 3 x 2 x 1

    8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1


Example 6
Example 6

In how many different ways can a race with six runners be completed? Assume there is no tie.


Solution
Solution

  • Six possible choices for first place

  • Five possible choices for second place

  • Four choices for third place and so on…

  • So, by the Fundamental Counting Principle, the number of different ways the race can be completed is:

    6 x 5 x 4 x 3 x 2 x 1 = 6! = 720


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