Jenn Jnen Jnne Ejnn Enjn Ennj Njen Njne Nejn Nenj Nnje Nnej

12

John Jhon Jonh Jhno Jnoh Ojhn Ojnh Ohnj Onjh Ohjn Etc…..

24

The double letter reduces the number of permutations

The two n’s can trade places and it be the same permutation as another.

To find the number of permutations with identical items, divide the total number of arrangements by the number of ways to arrange the identical items only.

How many ways could you arrange J.e.n1.n2?

4!

How many ways can you arrange n1.n2 ?

2!

4!/2! = 24/2 =12

How many permutations are there of the letters in the name Kristina

20160!2

!8

The number of permutations of a set of n items with a, identical items is:

!

!

a

n

Tiles used to surround a bathroom floor. There are 4 yellow, a blue, green, red and grey tile. How many patterns can be made?

There are 8 tiles in total, with 4 of them being identical

1680!4

!8

There will be times when some of the items in the set are alike.

For example; Suppose we are asked to find the number of ways of arranging the letters in the word

EXCELLENT.

So far we know that factorials will be used…

!nThe existence

of duplicate items changes

this slightly.!...!!

!

cba

n

a simply represents the # of times the first duplicate appears.

b simply represents the # of times the second duplicate appears.

And so on…

!...!!

!

cba

n

In this case we have;

3 E’s (a = 3)2 L’s (b = 2) !2!3

!9

30240Meaning that there are 30240 different ways to arrange

the letters in the word EXCELLENT.

Suppose we are asked to find the number of ways of arranging the letters in the word EXCELLENT.

How many permutations are there of the letters of WANNAN?

Things get a little different if you have more than one identical item.

In this case we have 3 n’s and 2 a’s

60!2!3

!6??

!6

You multiply all repeated items together before dividing

The number of permutations of a set of n items with a, b, c, which are different identical items is:

!...!!

!

cba

n

a) How many Permutations are possible of the letters of the word bookkeeper?

b) Barb has some clothes to display in her store window; 5 sweaters, 3 t-shirts, and 4 pairs of pants. How many ways can she arrange them?

15120024

!10

!3!2!2

!10

27720!4!3!5

!12

Page 245 #1-5, 7-12, 17a

QUIZ MONDAY!!!!