Systematic listing and the product rule for counting

Jodie makes a sandwich using one type of bread and one filling.

Complete the table of the 6 possible sandwiches they could make. One has been done for you.

What fraction of the possible sandwiches use Cheese?

Amy and Becky each make integers using three single digits. In any integer, digits may be repeated.

• Amy makes even integers with a first digit greater than 7

• Becky makes odd integers with a first digit that is non-zero.

They each make as many different integers as possible.

How many more integers than Amy does Becky make?

Callum is making pizza. He can choose three different toppings from

ham (H)

sweetcorn (S)

onion (O)

mushrooms (M).

List all the possible options for the three toppings. The first one has been done for you.

H S O

When x is divided by 2 the remainder is 1 When x is divided by 3 the remainder is 1 When x is divided by 4 the remainder is 1

Work out two possible values of x.

At a restaurant, vegan pizzas have two different toppings. The toppings are sweetcorn (S) mushrooms (M) peppers (P)

Complete the table to list all the possible pairs of toppings.

At the restaurant, dough balls can be ordered in small portions and large portions.

A group of people want to order exactly 44 dough balls.

Show how they can do this.

Fatima is choosing a 4-digit code. Each digit is a whole number from 0 to 9 She decides all her digits will be odd numbers no digits will be repeated.

How many different codes can she make?

A code is three letters, each of which is in the word LOGIC Vinny assumes that letters in the code may be used more than once. He works out how many possible codes there are. In fact, the first two letters are the same and the third is different.

How many of Vinny’s codes are not possible?

Here are the subjects available in year 12 at a school.

Students choose three subjects. They cannot choose more than one subject from a block. Lian decides to study Maths not to study Geography, Chemistry, Physics or ICT.

By listing, show that there are seven groups of three subjects that Lian could choose.

Work out how many 5-digit odd numbers can be made using these digits once each.

2 4 6 7 9 Do not list them.

A game has four cards labelled A, B, C and D. Ellie picks two of the cards at random. Complete the list of the six possible pairs of cards she could pick.

After school, Priya will

• go running (R)

• do her homework (H)

• play a video game (V).

Complete the list of the 6 possible orders in which she could do them.

Sue is working with 2-digit numbers. She multiplies the digits together to get an answer.

Write down a different 2-digit number that gives an answer of 18

Write down a 2-digit number that gives an answer of 0

Write down a 2-digit number that gives an answer greater than 70

In the grid, the product of each row, column and diagonal is 1

Complete the grid.

In a video game, players make their own character. They choose one of each from 8 faces 4 bodies 5 hairstyles.

How many different characters can be made?

Two characters are made at random. What is the probability that they are exactly the same?

Annie spends these amounts in four shops using £20 notes, £10 notes and £5 notes.

In each shop she pays the exact amount uses the smallest possible number of notes. Work out the total number of each note she uses.

Put the numbers 1, 2, 3, 4 and 6 into the circles so that each line of three numbers multiplies to 12 the total of the vertical line is one more than the total of the horizontal line. Use each number once.

Here are four number cards.

Use each card once to make this calculation correct.

Two of the cards are chosen at random.

List all the possible pairs of cards. Two have been done for you.

Write down the probability that the first card is an even number.

On a restaurant menu there are 22 main dishes, of which are gluten-free 7 rice dishes, which are all gluten-free 5 naan breads, of which 40% are gluten-free. This Meal Deal is on the menu. Choose one main dish, one rice dish and one naan bread How many of the possible Meal Deals are totally gluten-free?

A team of two players is picked from these people.

The team must have one female player and one male player. Complete this list to show all of the possible teams.

George buys some food for £16.55 He pays the exact amount with two notes and four coins. List the notes and coins.

Write down all the whole numbers that are between 20 and 50 and have a difference of 4 between their digits.

A building company employs 2 labourers 14 joiners 9 electricians 8 plumbers. For a job, the company needs one of each type of worker.

In how many ways can the company choose the four workers?

One labourer and two plumbers are on holiday. In how many ways can the company now choose the four workers?

Ali and Mel are making 3-digit codes. The digit 0 is not used. Ali only uses odd digits. Mel only uses even digits.

Ali can make x more codes than Mel. Assume that digits cannot be repeated. Work out the value of x.

In fact, digits can be repeated. What does this tell you about the actual value of x ? Tick one box.

Jim wants to buy 10 rolls of wallpaper. He sees these prices.

What is the cheapest price for 10 rolls?

A shop sells ice creams. Each ice cream has two scoops.

The possible flavours are vanilla (V), strawberry (S), chocolate (C) and mint (M). The two scoops can be the same flavour or different flavours.

List all the possible options for the two scoops.

In one hour the shop sells 180 scoops of ice cream. The number of scoops of each flavour is shown in the table.

Complete the pie chart to represent the data.

Here are five number cards.

Two of the five cards are picked at random. Work out the probability that the total of the two numbers is more than 30

Lena is at the gym.

She will use each of these pieces of equipment once. Rowing machine (R) Stepper (S) Treadmill (T) Bike (B) Lena will use the rowing machine first. List all the possible orders in which she could use the four pieces of equipment.

The table shows how long Lena spends on each piece of equipment.

Lena starts on the rowing machine at 1.50 pm She has a break for 4 minutes between pieces of equipment. What time does she finish on her last piece of equipment?

Sam, Carl and Erik share 40 sweets. Erik gets the largest share. What is the smallest possible number of sweets that Erik could get?

The numbers 1 to 12 are put in a grid. 2, 4, 5, 7, 10 and 12 are shown.

Each of the four sides of the grid must add up to 26 Complete the grid using the numbers 1, 3, 6, 8, 9 and 11

Nick has a 6-digit code. He remembers it as three 2-digit numbers. The first number is between 10 and 20 The second number is 3 times the first number. The third number is 5 times the first number. All six digits are different. Work out the code.

Jim has six banknotes. The value of each note is £5 or £10 or £20 He can make £20 with three notes. He can make £55 with four notes. He cannot make £25 with three notes. He cannot make £25 with four notes. List the six notes.

A music app has a shuffle play function. This means that songs are played in a random order without repeat.

Ruth puts 10 songs on shuffle play. One of them is her favourite song. Write down the probability that her favourite song plays first.

Ted puts songs A, B and C on shuffle play. List all the possible orders of songs A, B and C. One has been done for you.

A B C

Four positive whole numbers add up to 36 One of the numbers is a multiple of 7 The other three numbers are equal. Work out the result when the four numbers are multiplied.

14^{Do not write}_{outside thebox}

Here are two methods to make a 4-digit code. Codes can have repeated digits.

Method A For the first two digits use an odd number between 30 and 100 For the last two digits use a multiple of 11

Method B Use four digits in the order even odd even odd Do not use the digit zero

Which method gives the greater number of possible codes? You must show your working.

In a game, three stars are hidden at random. Each star is behind a different square on this board.

A square is chosen at random. What is the probability that there is a star behind it?

In one game, the stars are behind three consecutive squares. The squares are in one row or one column. One of the squares is E2 Write down all the possible pairs for the other two squares.

Meal Deal Choose one sandwich, one drink and one snack There are 7 different sandwiches 5 different drinks and 3 different snacks.

How many different Meal Deal combinations are there?

Two of the sandwiches have cheese in them. Three of the drinks are fizzy. Eva picks a Meal Deal at random. Work out the probability that the sandwich has cheese in it and the drink is fizzy. Give your answer as a fraction.

Eve has these coins.

Ola has these coins.

Eve gives three of her coins to Ola. Now, Ola has the same amount of money as Eve.

Which coins does Eve give to Ola?

Gina makes a sandwich using bread (B) or a roll (R) and ham (H) or cheese (C) and salad (S) or pickle (P)

List all the possible types of sandwich Gina could make. One has been done for you.

B H S

What fraction of the possible types of sandwich have cheese and pickle?

A farmer has 580 eggs to put into boxes. The boxes come in three sizes.

at least 10 boxes of 20 eggs at least 15 boxes of 12 eggs at least 25 boxes of 6 eggs. The farmer fills 54 boxes with the 580 eggs. Show how he does this.

boxes of 20 eggs

boxes of 12 eggs

Bottles of drink are for sale at three shops. The normal price of a bottle is the same at each shop.

What is the cheapest way to buy exactly 8 bottles? You can buy from more than one shop. You must show your working.

A code has 4 digits. Each digit is a number from 0 to 9 Digits may be repeated. The code starts 5 4 1

Amy knows the last digit is odd but not 7 She chooses a different odd number at random. What is the probability that she chooses the correct number?

The 4-digit code is changed to an even number. The first digit is 3 How many possible codes are there?

To make one cheese sandwich, Gina uses one bread roll and two cheese slices.

She is going to buy enough packs to have exactly twice as many cheese slices as bread rolls make more than 100 cheese sandwiches. Work out the least amount she can spend.

Twelve cards numbered 1 to 12 are put into six pairs. Each pair has a total. Complete the table to show the pairs and their totals.

The value of x can be 2 or 5 The value of y can be 3 or 12

List the possible values of xy

Work out the least possible value of You must show your working.