Systematic listing and the product rule for counting

There are 15 dogs in a dog show. One dog is awarded first prize. A different dog is awarded second prize. There are 12 cats in a cat show. One cat is awarded first prize. A different cat is awarded second prize. Work out how many different ways these four prizes can be awarded.

Here is a menu.

Asma can choose one drink and one cake. Write down all the possible combinations Asma can choose.

There are 10 football teams in a league. Each team plays every other team 4 times. Work out the total number of games played.

Anita throws a coin 3 times. Each time the coin can land on heads (H) or tails (T). List all the possible outcomes.

There are 30 students in a class. A teacher is going to choose at random 2 of the students. Work out the number of different pairs of students that the teacher can choose.

Harris is buying a shirt and a tie. He has a choice of three colours of shirt and a choice of three styles of tie.

Harris is going to choose one shirt and one tie. List all the possible combinations Harris can choose.

Zia has to set a 4‑digit security passcode on her phone. Each digit of the passcode is a number from 1 to 9 She can use each number more than once. Zia tells her friend Amber that the first digit is a cube number the second digit is a prime number the third digit is greater than 6 the fourth digit is an odd number. The diagram shows one possible 4‑digit passcode.

Amber is going to have one attempt at guessing Zia’s passcode. Work out the probability that Amber guesses Zia’s passcode on the first attempt.

The menu in a restaurant has starters, main courses and desserts. There are 5 starters. There are 12 main courses. There are x desserts. There are 420 different ways to choose one starter, one main course and one dessert. Work out the value of x.

x and y are integers such that 3 < x < 8 4 < y < 10 and x + y = 14 Find all the possible values of x.

A pet shop has 7 guppy fish 13 tetra fish 5 angel fish. David is going to choose one of the following combinations of fish a guppy fish and an angel fish or a tetra fish and an angel fish or a guppy fish, a tetra fish and an angel fish. Show that there are 555 different ways for David to choose his fish.

Rayheem has 16 shirts 5 pairs of jeans 3 jackets Rayheem chooses an outfit to wear. An outfit is 1 shirt, 1 pair of jeans and 1 jacket. Work out how many different outfits Rayheem can choose.

In a school there are 16 teachers and 220 students. Of these students 120 are girls and 100 are boys. One teacher, one girl and one boy are going to be chosen to represent the school. Work out the number of different ways there are to choose one teacher, one girl and one boy.

There are three cards in bag A and two cards in bag B. There is a letter on each card.

James takes a card from bag A and then a card from bag B. List all the possible outcomes.

Lucy uses a code to open a lock. The code is a letter followed by a 2-digit number. The letter is L or U. The number is a prime number between 20 and 30 Write down all the possibilities for Lucy’s code.

Jack is in a restaurant. There are 5 starters, 8 main courses and some desserts on the menu. Jack is going to choose one starter, one main course and one dessert. He says there are 240 ways that he can choose his starter, his main course and his dessert. Could Jack be correct? You must show how you get your answer.

Mason throws a coin 3 times. The outcome of each throw is either Heads or Tails.

List all the possible outcomes of the 3 throws.

Sadia is going to buy a new car. For the car, she can choose one body colour, one roof colour and one wheel type.

She can choose from 19 different body colours 25 different wheel types

The total number of ways Sadia can choose the body colour and the roof colour and the wheel type is 3325

Work out the number of different roof colours that Sadia can choose from.

There are three dials on a combination lock. Each dial can be set to one of the numbers 1, 2, 3, 4, 5 The three digit number 553 is one way the dials can be set, as shown in the diagram.

Work out the number of different three digit numbers that can be set for the combination lock.

How many of the possible three digit numbers have three different digits?

In a restaurant there are 9 starter dishes 15 main dishes 8 dessert dishes

Janet is going to choose one of the following combinations for her meal.

a starter dish and a main dish or a main dish and a dessert dish or a starter dish, a main dish and a dessert dish

Show that there are 1335 different ways to choose the meal.

Mohsin, Yusuf and Luke are going to play a game. At the end of the game, one of them will be in First place, one of them will be in Second place and one of them will be in Third place.

Use the table below to list all the possible outcomes of the game.

There are 16 hockey teams in a league. Each team played two matches against each of the other teams.

Work out the total number of matches played.

Tracey is going to choose a main course and a dessert in a cafe. She can choose from 8 main courses and 7 desserts.

Tracey says that to work out the number of different ways of choosing a main course and a dessert you add 8 and 7

Is Tracey correct? You must give a reason for your answer.

12 teams play in a competition. Each team plays each other team exactly once.

Work out the total number of games played.

Priti is going to have a meal. She can choose one starter and one main course from the menu.

Write down all the possible combinations Priti can choose.

Jeff is choosing a shrub and a rose tree for his garden. At the garden centre there are 17 different types of shrubs and some rose trees.

Jeff says,

“There are 215 different ways to choose one shrub and one rose tree.”

Could Jeff be correct? You must show how you get your answer.