AN INTRODUCTION TO PERMUTATIONS

Slideshow 60, Mathematics

Mr Richard Sasaki, Room 307

OBJECTIVES

• Understand the meaning of a permutation

• Recall how to calculate permutations with repetition (replacement)

• Learn how to calculate permutations without repetition (without replacement)

PERMUTATIONS

What is a permutation?

A permutation is an arrangement of objects (numbers, letters, words etc…) in a specific order.Let’s list the possible ways we can pick two numbers from 1, 2 and 3 where repetition is okay.

1, 11, 21, 3

2, 12, 22, 3

3, 13, 23, 3

If is the # of possibilities and is the # of times we choose, there are…

permutations.𝑛𝑟

PERMUTATIONS WITH REPETITION

We have seen this before. If we use this formula, we can easily calculate the number of permutations possible.ExampleThree meetings will take place next week and are possible to happen on any day of the week. It is possible for one, two or all three meetings to take place on a single day. How many possible ways are there for the meetings to take place?

𝑛=7 ,𝑟=3 ,𝑛𝑟⇒73 ways

368

81

Months have differing numbers of days. .

206

364

PERMUTATIONS WITHOUT REPETITION

So, permutations with repetition are simple! But without repetition (where a value can’t appear twice) is more complicated. Let’s see the difference.Example (repetition)

A four digit number is made where zero is allowed in all four positions. How many permutations are there?

Example (without repetition)

; four times

𝑛𝑟¿104¿10,000

?? ??

possible9

possible8

possible

¿10 ∙9 ∙8 ∙77

possible

¿5040

…?

Does this pattern remind you of anything we learned briefly at the start of this year?

¿10 ∙9 ∙8 ∙7 ∙6 ∙5 ∙4 ∙3 ∙2 ∙1¿10 !

Factorials!

How can we represent in terms of factorials?

10 !¿¿6 ∙5 ∙4 ∙3 ∙2∙1¿10 !6 !

How about the general case…with and ?

PERMUTATIONS WITHOUT REPETITION

In our example, we have 10 choices for each digit and there are 4 so…

𝑛=¿ 𝑟=¿10 410 !6 !

The top number is obviously 𝑛 !_____But where does the bottom number come from? (____ = 6!…?)

We clearly did 10 – 4, right? ()𝑛−𝑟

But we got 6!, not 6…so…

()!

is for permutations without repetition where is the # of possibilities and is the # of times we choose.

PERMUTATIONS WITHOUT REPETITION

Example

How many ways can be ordered where only 2 letters are used but each can’t be used twice?

𝑛=¿ 𝑟=¿4 2𝑛 !

(𝑛−𝑟 ) !¿4 !

(4−2 ) !¿4 !2 !

¿4 ∙3 ∙2 ∙12 ∙1

¿242¿12

12 ways (-

)

ANSWERS

120

240

210

72

81

ANSWERS　 (PART 3)

6421649256

241204224

With repetition: , Without repetition:

ANSWERS　 (PART 4)

6𝑛=3 ,𝑟=3

h𝑤𝑖𝑡 𝑜𝑢𝑡

𝑛 !36

30

12

48

ANSWERS　 (PART 5)

220720

Because we can’t pick more options than there are as one is removed each pick.Or…factorials need to be positive, (n – r)! .

9P8

870Crazy: n = 5, r = 5 Hello has ‘l’ twice (indistinguishable) so it will have less than 5!.

ANSWERS　 (PART 6)

64720

67

You can’t get 7 different outcomes from a die.

4 !=24156

49 !45 !