1.7 Permutations & Combinations

Three letters are chosen at random from the letters in the word AIRFIELD.  

Find the number of ways that the selection may contain

Write down the number of arrangements of three letters chosen at random from the word AIRFIELD that have exactly one letter I.

In a maths test students are required to answer four out of seven questions.

Find the number of ways in which the questions can be chosen if there are no restrictions.

Find the number of ways in which the questions can be chosen if the last question is compulsory.

Find the number of ways in which the questions can be chosen if the students must do at least 1 of the last two questions.

A farm has a new litter of kittens.  Two of the kittens are classed as mostly white, four are mostly black, and five are classed as black and white mixed.

Five of the kittens are selected at random.

Find the number of ways in which the selection might contain:

Find the number of ways in which the selection might contain at least two kittens that are classed as mostly black.

A pool table has fifteen different balls including the black ball.  

Given that the black ball is the last to be potted and the rest of the balls are potted one at a time in a random order, in how many ways can the fifteen balls be potted?

The other fourteen balls consist of seven pairs of different coloured balls.  One of each pair has a stripe across it and the other has a spot on it.  

Given that the black ball is still the last to be potted, in how many ways can the fifteen balls be potted one at a time if

Ms Aiba has twelve different maths textbooks on her classroom bookshelf.  Five of them are Statistics textbooks and the other seven are Pure Mathematics textbooks.  Determine the number of different ways that the books can be arranged on the shelf if

Nine shirts for a baseball team are numbered from 1 to 9.  Five players are allowed to take one shirt each.

Find the number of ways this can be done if 

Riley is going on holiday and is allowed to bring along four of his toys.  At home he has nine different plastic dinosaurs, six different toy cars, and five different wooden reptiles.

How many different selections of his toys can he make if he chooses at least one of each type of toy?

Riley can’t decide so he persuades his parents to allow him to bring along five toys instead.  

Given that he brings more plastic dinosaurs than any other type of toy, how many different selections can he make now?

Two parents, Julie and Malcolm, have 15 household chores to be given to their five children, Biddy, Gus, Mandy, Charlie and Claire. 

Find the number of ways in which the 15 household chores can be distributed if Biddy is to do 5 of them, Gus is to do 4 of them, Mandy is to do 3 of them, Charlie is to do 2 of them and Claire is to 1 of them. 

A team of 16 is to be split up in order to complete a number of tasks.

Find the number of ways the team can be divided into

A quarter of the team are unable to help with the tasks due to COVID-19 infections.

Find the number of ways that the remaining team members can be divided into

You have a set of four different coloured flashlights that you use to send messages to your friend who lives across from you on the same street.

(For purposes of this question, a ‘message’ is considered to be a single sequence of one or more coloured lights, displayed one after the other.)

Find the number of different messages you can send using two different coloured flashlights displayed one after the other.

Find the number of different messages you can send using three different coloured flashlights displayed one after the other.

Find the total number of different messages you could send using your four flashlights, under the condition that all the colours in a given message must be different from each other.

In this question give your answers in the form , where  and 

John is a builder who has divided his day’s work into 15 separate tasks.  Two of these tasks John considers ‘physically demanding’ because one includes moving bricks and the other includes concrete mixing. 

Find the number of ways John can complete the tasks given that one ‘physically demanding’ task is done at the start of the day and the other ‘physically demanding’ task is done at the end of the day.

Find the number of ways John can complete the tasks given that the two physically demanding tasks are not done consecutively.

A horse-riding club uses 8 small fields arranged in two rows of four to allow its horses to graze.

The riding club has 8 horses in total, but three of the horses – named Angel, Blue and Chance – get agitated if they are placed too close to each other for too long. 

For each of the following sets of conditions, find the number of possible ways of placing the 8 horses in the 8 fields. 

Each field may contain up to 6 horses. Angel, Blue and Chance, however, must each be in a separate field.

Each field may contain at most one horse. Additionally Angel must be in one of the corner fields, and Angel, Blue and Chance must not be placed in fields which share any boundaries (it is okay if the fields meet at a corner, but they may not share a side).

Consider the ten letters in the word QUARANTINE. 

Find the number of ways that four letters can be chosen at random from the word QUARANTINE if the selection may contain

Find the number of distinct ways four out of the ten letters of the word QUARANTINE can be arranged if in each case all four letters are different from each other.

Consider the eight letters in the word CAMPFIRE. 

Find the number of ways in which the eight letters may be arranged if

Find the number of ways in which the eight letters in the word CAMPFIRE may be arranged, given that at least two of the vowels must be next to each other

Ahmed is playing a game with the following nine cards:

Ahmed arranges the cards to form a 9-digit number. Find how many different 9-digit numbers can be made such that the number is a multiple of five and the circular cards are not all together.

Ahmed chooses one triangular card, one circular card and one rectangular card at random and arranges them to make a 3-digit code. Find the number of different 3-digit codes Ahmed could make.

Jonni always struggles to decide which combination of fillings, sides and sauces he wants to put in his sandwich at his favourite sandwich shop. The available options are listed below: 

To avoid having to make up his mind, each day Jonni instead sets himself a rule for how many of each option he will put in his sandwich, and then uses an app he has designed to choose one sandwich at random from among those that his rule allows.  Note that choosing the same filling, side or sauce more than once is never allowed.

If Jonni’s rule on a given day is that he will have one filling, three sides and one type of sauce in his sandwich, find the probability that Jonni has either steak or chicken in his sandwich.

If Jonni’s rule on a given day is that he will have two fillings, one side and three types of sauce in his sandwich, find the probability that he gets either tuna with tomato, or jalapenos with ketchup and mayo, as a part of his sandwich choice.

Twenty-six people are travelling on an airplane. Seven of them are businesspeople on their way to a facilities management conference. Three of them are spies on their way to take up positions as ‘consular officials’ in an unnamed embassy. The other sixteen are cryptozoologists on their way to follow up some clues regarding a recent unicorn sighting. These passengers have been randomly allocated the seats from 26A to 29E inclusive, as depicted in the diagram below. 

Find the number of ways that all twenty-six people could be seated on the plane if the three spies are all sat in the back row and the seven businesspeople are all in seats by a window.

Given that each of the seven window seats has a cryptozoologist in it, find the probability that the three spies are not all sat together in one of the groups of three seats in between the two aisles.

The following four family groups, consisting of 6 adults and 12 children in total, are all going to the cinema together: 

• Mr and Mrs Mitchell and their two children 

• Mrs and Mrs Lee and their four children 

• Mr Kim and his three children 

• Ms Miller and her three children 

Find the number of different ways that all 18 people can sit in a row of 18 seats if

If instead the 18 people are sat randomly in three rows of six, what is the probability that all of the adults will be sat in the back row?

The diagram below shows part of a train carriage. There are twenty-eight seats consisting of twenty normal seats, S, and eight table seats, T. In this part of the train carriage there is a group of twelve schoolchildren and two teachers, along with a married couple, four businessmen, and three backpackers.

Find the number of ways that the passengers could be arranged in this part of the train carriage if

Given that everybody is arranged completely at random within this part of the train carriage, find the probability that

A total of n tasks are to be shared among m people, where each of the people is uniquely identified by a number between 1 and .  Person 1 is to receive  tasks, Person 2 is to receive  tasks, and so on, with the integers  being such that        and . 

Show that the number of different ways in which the tasks can be assigned according to the above rules is