Unit 1 - Number and Algebra
Get Ready
Questions
In how many ways can the letters A, B, C, D and E be arranged?
Solutions
Notes
The number of ways of arranging \(n\) distinct objects in a row is \(n!\)
The number of ways both \(A\) and \(B\) can happen (provided \(A\) and \(B\) are independent) is given by \(n\left(A\right)\times n\left(B\right)\).
For example, if a set menu in a restaurant has \(3\) choices for a starter (A, B, C) and \(4\) choices for a main course (W, X, Y, Z), then there are a total of \(3\times 4=12\) possible meals that can be ordered.
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For each of the \(3\) available starters, a customer could choose any of the \(4\) main courses. So all the options are$$\begin{matrix}AW&AX&AY&AZ\\BW&BX&BY&BZ\\CW&CX&CY&CZ\end{matrix}$$
The number of ways of either \(A\) or \(B\) can happen (provided \(A\) and \(B\) are mutually exclusive) is given by \(n\left(A\right)+n\left(B\right)\).
For example, if there are \(5\) films on Netflix that I want to watch and \(7\) films on Disney+ I want to watch, then there is a total of \(5+7=12\) possible choices for the film I watch this evening.
Examples and Your Turns
Example
A menu has \(3\) starters, \(5\) mains and \(2\) desserts.
a) Anne would like to order a starter, a main and a dessert. How many different choices could she make?
b) Bill would like to order a main and either a starter or a dessert (but not both). How many different choices could he make?
Your Turn
An examination is made up of three sections, A, B and C. Section A has \(4\) questions, Section B has \(8\) questions, and Section C has \(3\) questions.
a) How many different ways are there to choose questions if you are required to select one question from each section?
b) How many different ways are there to choose questions if you are required to choose one question from either section A or Section B (but not both) and then one question from Section C.
Example
A seven digit number is formed using the digits 1-7 once each.
a) How many different numbers are there?
b) How many such numbers are even?
Your Turn
a) How many permutations are there of the letters in the word SQUARE?
b) How many of these start with two consecutive vowels?
Notes
A permutation of a collection of objects is an arrangement of the objects.
The words STAR and ARTS are two permutations of the letters A, R, S and T.
In total there are \(4!=24\) different permutations of these four letters (most of them are not words).
Now let us consider what happens if we want permutations that don’t use all the letters.
For example, the word A is a possible permutation when we take one letter from the four available.
How many different permutations are there when we take one letter from the set?
How many different permutations are there when we take \(4\) letters at a time from the set of all \(26\) letters in the alphabet?
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There are four permutations we can have: A, R, S or T.
When we take two letters at a time from the set we can get words such as AT or AS.
How many different permutations are there when we take two letters at a time from the set?
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Listing systematically we find the following possible permutations$$\begin{matrix}AR&RA&SA&TA\\AS&RS&SR&TR\\AT&RT&ST&TS\end{matrix}$$So in total there are 12 permutations.
We can calculate this using the calculation \(4\times 3=12\) as there are \(4\) possible starting letters, and for each of these there are \(3\) possible second letters.
When we take three letters at a time from the set we can get words such as SAT or TAR.
How many different permutations are there when we take three letters at a time from the set?
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Listing systematically we find the following possible permutations$$\begin{matrix}ARS&RAS&SAR&TAR\\ASR&RSA&SRA&TRA\\ART&RAT&SAT&TAS\\ATR&RTA&STA&TSA\\AST&RST&SRT&TRS\\ATS&RTS&STR&TSR\end{matrix}$$So in total there are 24 permutations.
We can calculate this using the calculation \(4\times 3\times 2=24\) as there are \(4\) possible starting letters, and for each of these there are \(3\) possible letters for the second letter and then \(2\) possible letters left for the third letter.
This process can take a long time if the original set is large, and we are selecting a large subset of letter.
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Listing systematically here would be impossible as there are so many possible permutations.
Instead we consider the process of thinking about how many options there are for each letter.
For the first letter there are \(26\) choices.
For the second letter there are \(25\) remaining letters to choose from.
For the third letter there are \(24\) remaining letters to choose from.
For the final letter there are \(23\) remaining letters to choose from.
So in total there are$$26\times 25\times 24\times 23=\frac{26!}{22!}=358 800$$possible permutations.
From this we can deduce a formula.
The number of permutations of \(r\) objects from a list of \(n\) distinct objects is given by$${}^{n} \mathrm{ P }_{r}=\frac{n!}{\left(n-r\right)!}$$
Examples and Your Turns
Example
A class of \(28\) students must select a committee consisting of a class representative, a treasurer, a secretary and a football captain. The same person cannot hold two posts. In how many different ways can the four posts be filled?
Your Turn
If a chess association has \(16\) teams, in how many ways could the top \(8\) positions be filled in the competition ladder?
Your Turn
There are \(20\) different chocolates in a box. In how many different orders could I eat 3 chocolates?
Your Turn
There are \(14\) books that Tom would like to borrow from the library, but he is only allowed to check out \(5\) books at a time. In how many ways could Tom rank his top 5 books?
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.