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Arrangments of items


A permutation is a particular way that a group of objects could be selected and arranged. For example, If I have a red dot, blue dot and green dot; then I could choose any two of them and end up with the following options.

The idea of counting how many possible different combinations there are is part of a mathematical topic called Counting Techniques. For small sets like the one above, it's quite simple to write down all the choices and add them up. But for larger sets, like imagine we had $8$8 colours and wanted to see how many different combinations of $5$5 colours there are - we need a more efficient method of counting.

You'll notice that in the example above with the coloured spots, a red and green spot is a different permutation to a green and red spot. We say that for a permutation, that the order is important. That is, we call the red and green different to a green and a red.


Diagramatic Calculations

Using a diagram can help us work out the number of permutations a set of objects has.

For example, consider the set of letters $A,B,C,D$A,B,C,D and $E$E. We want to see how many different ways we can select and arrange a set of $3$3 letters.

A diagramatic calculation starts with drawing boxes (or circles, or triangles, or any other shape you can write a number inside of), where the number of the boxes is equal to the number of objects we are selecting. In this case we are selecting $3$3 letters each time, so we have $3$3 boxes.

Inside the first box, we write down how many possible choices we have for this box. We have $5$5 letters to choose from, so we have $5$5 options. Either $A,B,C,D$A,B,C,D or $E$E.

Inside the second box, we write down how many possible choices we have for this box. Because we had $5$5 to start with, we will have selected one already, so there are $4$4 options left for this next box.

Finally, inside the third box, how many possible letters will we have left to pick from? Only $3$3.

To calculate how many possible options we have we multiply these together.

So there are $60$60 possible permuatations (or arrangments) for selecting $3$3 objects from the $5$5.


Worked examples


There are $4$4 people who are lining up outside a store for an amazing annual sale. Fred, Ginger, William and Asteria. How many different arrangements are there for the four people in the line?

Think: We know that to fill the first position in the line we have $4$4 people to pick from. This leaves us with only $3$3 for the next position, and then $2$2 for the next and $1$1 for the last.

Do: This results in $4\times3\times2\times1=24$4×3×2×1=24.



Determine the number of ways the letters of the word $HAPPY$HAPPY can be arranged in a line.

Think: There are $5$5 letters in the word $HAPPY$HAPPY. We want to keep in mind that the letter $P$P appears twice. This means that unlike arranging $5$5 unique letters, when we swap the letter $P$P with the other $P$P, we don't get a new arrangement.

Do: If $HAPPY$HAPPY was made up of $5$5 unique letters, then the result would be $5\times4\times3\times2\times1=120$5×4×3×2×1=120.

But this counts each permutation twice since it swaps position of the letter $P$P with the other $P$P. So we want to halve the resulting value.

$\text{Number of arrangements }$Number of arrangements $=$= $\frac{120}{2}$1202
  $=$= $60$60


Practice questions

question 1

Lucy's ATM pin is a $4$4-digit number.

She remembers the first digit, but cannot remember the next three digits that follow.

How many different possibilities are there?



At a gathering of political leaders, there are $3$3 delegates from European countries, $5$5 delegates from African countries, and $2$2 delegates from South American countries.

During question time, they are to be seated in a row on stage, however the South American delegates need to leave early and so are seated together to the right of the other delegates.

In how many ways can the delegates be seated?



In a certain code, the digits $0$0 and $1$1 are placed together to form a string and each string represents a word. For example, $11010$11010 is a string.

How many of these strings can be created using three $0$0s and four $1$1s?


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