Fundamental Counting Principle and Tree Diagrams

Counting Principle L1

By Henry Phan

Multiple Choice

Use the given tree diagram to determine stage 1

• Fried Tortellini, Marinated Roast Turkey

• Fried Tortellini, Turkey Pinwheels, or Sheet Pan Nachos

• Fried Ice Cream Bars or Irish Cream-Topped Brownie

• Curry Roast Chicken or Marinated Roast Turkey

Multiple Choice

Use the given tree diagram to determine stage 2

• Fried Tortellini, Turkey Pinwheels, or Sheet Pan Nachos

• Fried Tortellini, Turkey Pinwheels, or Sheet Pan Nachos

• Fried Ice Cream Bars or Irish Cream-Topped Brownie

• Curry Roast Chicken or Marinated Roast Turkey

Fill in the Blank

Determine how many combinations of food choices are possible? (input just a number)

Multiple Choice

Use the given tree diagram to determine stage 3

• Fried Tortellini, Turkey Pinwheels, or Sheet Pan Nachos

• Fried Tortellini, Turkey Pinwheels, or Sheet Pan Nachos

• Fried Ice Cream Bars or Irish Cream-Topped Brownie

• Curry Roast Chicken or Marinated Roast Turkey

Fill in the Blank

What are the choices that a guest will select Turkey Pinwheels dishes

​The fundamental counting principle states that if there are p ways to do one thing, and q ways to do another thing, then there are p × q ways to do both things. possible outcomes of the experiment. The counting principle can be extended to situations where you have more than 2 choices.

​Stage 1:  # choices

Stage 2:  # choices

Stage 3:  # choices

……

Stage last: # choices

​Example:

​Justine is a wedding coordinator. She is selecting the menu options for the reception. First, she will give the guests a choice of Salad or Soup for the first course. Then, she will allow them to pick from a chicken, steak, or a vegetarian for their main course.  Finally, she will allow the guests to choose bride’s vanilla cake or groom’s chocolate cake.  Determine how many combinations of food choices are possible?

At a Used Car Dealership, the coming new vehicles are  posted this morning today with a particular model comes in 2 body types (sedan or hatchback) 5 different colors  (Night Black, Fire Red, Lime Green, Navy Blue, or Icy Blue). All these cars come in 3 models: GL (standard model), SS (sport model with bigger engine), or SL (luxury model with leather seats).  How many combinations of car versions of the models can be created from these options?​

Multiple Choice

John is a party coordinator.  He is selecting the menu options for the reception. First, he will give the guests a choice of Bread Burns, Bread Rolls or Rice for the first course. Then, he will allow them to pick from chicken, shrimp, turkey, or steak for their main course. Finally, he will allow the guests to choose for ice-cream: Blueberry goat cheese, or green tea. How many choices of dishes can be made for those options?

• Number choices

  3 x 4 x 2

• Number choices

  5 x 4 x 2

• Number choices

  3 x 4 x 5

• Number choices

  5 x 4 x 5

Multiple Choice

At a New Car Dealership,  a particular model comes in some trim levels  (CX, DX, EX, and Si).  The same model comes in a multiple of colors  (Silver Ice, Charcoat Grey, Shady Blue, Sandy Red Gold, Sassy Pink, and Apple Green).  The model of car also has a couple of interior options (Grey Cloth, Tan Cloth, Black Leather).  How many different versions of this model can be created from these options?

• Number Versions:

  8 x 13 x 6

• Number Versions:

  4 x 13 x 6

• Number Versions:

  4 x 6 x 6

• Number Versions:

  4 x 6 x 3

​At a Used Car Dealership, the vehicles for sale under $6000  for today are  posted this morning with a particular model comes in a few models (SUV, sedan, and van) several difference brands (Toyota, Honda, Nissan, Mitsubishi, and Mazda).  All these Japanese cars come in 4 different trim levels  (DX, LX, EX, and SE).  The same model comes in 3 different colors (Night Black, Pearl White, and Forest Green). How many combinations of car versions of the models can be created from these options?

​Key Terms

q Restriction: choosing among numbers (example 2 – 5 means 2, 3, 4, and 5)

qNo restriction = choices from 0 – 9 = 10 choices

qNo repeat:

§1st choices 0 – 9: 10 choices

§2nd choices 1 – 9 = 9 choices

§ 3rd choice 2 – 9 = 8 choices

§4th choice 3 – 9 = 7 choices

……… and so on

How to determine the restricted numbers?

​Restricted number: (last digit – fist digit + 1)

​1)How many numbers are from 1 through 9?

​2) How many numbers are from 3 through 7?

Fill in the Blank

1)How many numbers are from 2 through 8?

Fill in the Blank

1)How many numbers are from 4 through 7?

Passcode, phone, and SS Cards

A four-digit social security number is of the form ABCD. The digit ‘A’ is restricted to a number 2 through 8. The digits B and C can be any number, but they cannot repeat.  The digits D has no restriction.  How many combinations of four-digit social security numbers are possible with these restrictions?

Subject | Subject

Some text here about the topic of discussion

Passcode, phone, and SS Cards

A six-digit passcode number is of the form  ABCDEF.  The digit ‘A’ is restricted to any number between  3 and 9. The digits B and C are restricted to any number between 2 and 9.  The digits D,E, and F can be any number, but they cannot repeat. .  How many combination numbers are possible with these restrictions to create six-digit passcode?

Subject | Subject

Some text here about the topic of discussion

Multiple Choice

A four-digit passcode number is of the form ABCD. The digit ‘A’ is restricted to a number 1 through 9. The digits B and C can be any number, but they cannot repeat.  The digits D has no restriction.  How many combinations of four-digit passcode numbers are possible with these restrictions?

• 10 x 10 x 10 x 10

• Passcode numbers:

  9 x 10 x 9 x 10

• 10 x 10 x 9 x 10

• 10 x 9 x 8 x 7

Multiple Choice

You want to create a six-digit passcode number which is of the form ABCDEF for your iPad. The digit ‘A’ is restricted to a number 1 through 9. The digits B, C, D and E can be any number, but they cannot repeat.  The digits F has no restriction.  How many combinations of six-digit passcode numbers are possible with these restrictions?

• Combination

  numbers:

  10 x 9 x 8 x 7 x 6 x5

• Combination

  numbers:

  9 x 10 x 10 x 10 x 10 x 10

• Combination numbers:

  9 x 10 x 9 x 8 x7 x 10

• Combination numbers:

  9 x 10 x 9 x 8 x7 x6