Lesson Worksheet: Counting Outcomes Mathematics

In this worksheet, we will practice counting outcomes using tree diagrams and the fundamental counting principle.


Michel, Kayla, and ChiWei are playing a game, where one of them needs to be a sheriff and one needs to be an outlaw. They write each of their names on a piece of paper and place them in a bowl. If two names are picked at random where the first will be a sheriff and the second will be an outlaw how many different ways are there?


A pizza parlor offers customizable pizzas. Customers can choose one of two options for the base, cheese, and sauce and then add an optional topping.

BaseCheeseSauceTopping (at Most One)
Thin crust or deep panCheddar or mozzarellaTomato or barbecuePepperoni, pineapple, or mushroom

Using tree diagrams, or otherwise, represent all the possible pizza combinations and determine how many different pizzas there are.


An ice cream store offers 3 different cup sizes and 14 flavors. How many ways are there to buy a single flavor of ice cream?


A coin is flipped 3 times. Use the following tree diagram to determine the event 𝐴 that includes all of the outcomes where no more than two tails are flipped.

  • A𝐴={(,,),(,,),(,,),(,,),(,,),(,,),(,,)}HHHHHTHTHHTTTHHTHTTTH
  • B𝐴={(,,),(,,),(,,)}HTTTHTTTH
  • C𝐴={(,,),(,,),(,,),(,,)}HTTTHTTTHTTT
  • D𝐴={(,,),(,,),(,,),(,,)}HHHHHTHTHTHH


A coin is flipped three consecutive times. Which of the following represents the event of getting a head on the first flip?

  • A𝐴={(,,),(,,),(,,),(,,)}HHHHHTHTHHTT
  • B𝐴={(,,),(,,),(,,)}HHHHHTHTH
  • C𝐴={(,,),(,,),(,,),(,,)}THHTHTTTHTTT
  • D𝐴={(,,)}HHH


A cafe offers a choice of 20 meals and 9 beverages. In how many different ways can a person choose a meal and a beverage?


A restaurant serves 2 types of pie, 4 types of salad, and 3 types of drink. How many different meals can the restaurant offer if a meal includes one pie, one salad, and one drink?


In how many ways can a 5-digit code be formed using the numbers 1 to 9? Note, the code can have repeated digits.


A school gives three prizes for excellence. The short lists for the prizes contain 9 students, 7 students, and 6 students. In how many ways can the prizes be distributed?


A car dealership offers 73 different models of cars and 14 different colors. Determine the number of ways someone can pick a car in a single color.

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