Permutations

Permutation is an arrangement of n different

objects with consideration given to the order of the

objects.

Notice, ORDER MATTERS

To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

Permutations

The number of ways to arrange the letters ABC: ____ ____ ____

Number of choices for first blank? 3 ____ ____

3 2 ___ Number of choices for second blank?

Number of choices for third blank? 3 2 1

3*2*1 = 6 3! = 3*2*1 = 6

ABC ACB BAC BCA CAB CBA

In general, the # of permutations of n objects is taken n at a time is:

Try…

1. 8 P 8 = 8! =

2. 5 P 5 = 5! =

3. 4 P 4 = 4! =

Example 1.

In how many ways can a boy arrange his 5 different toys in a row?

n = 5

5P5 = 5!

= 120

Example 2

How many different ways can 12 skiers in the Olympic finals finish the competition? (if there are no ties)

12P12 = 12!

=12*11*10*9*8*7*6*5*4*3*2*1

= 479,001,600 different ways

Example 3.

Six people are about to enter a cave in a single

file. In how many ways could they arrange

themselves in a row to go through the entrance?

6P6 = 6!

n= 6

= 720 ways

The number of Permutation of n different objects taken r at a time is denoted and defined, as follows:

Must try…

Example 1

Find the number of permutations using the 4 different letters a, b, c and d, if they are taken 2 at a time.

4 different objects means n = 4 and taking 2 at a time

means r = 2

12 permutations

Example 2

A permutation lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock permutations are possible assuming no number is repeated?

2436028*29*30)!330(

!30330

27!

30! p

Example 3.

Fifteen cars enter a race. In how many different

ways could trophies for the first, second and third

place be awarded?

n = 15, r = 3

waysp 730,213*14*1512!

15!

)!315(

!15315

Circular permutation When objects are arranged in a circle, the counting technique used to find the number of permutations is called circular permutation.

To determine the number of circular permutations, we shall consider one object fixed and calculate the number of arrangements based on the remaining number of objects left.

The number of circular permutations of n different objects is defined in symbols by:

Example 1

If 6 persons are to be seated in a round table with 6 chairs, how many ways can they be seated?

n = 6

= ( 6 – 1 )!

= 5!

= 120 ways

How many ways can 6 ladies be seated in a circular table

such that 2 of the ladies must always sit beside each

other?

(n – 1)! nPr = ( 5 – 1)! 2P2

4! X 2!

48 ways

Example 2

Permutation of n with alike objects Another type of permutation wherein the n, some of the r objects are alike, is known as permutation with alike things. This type of permutation is defined as:

Example 1 How many permutations are there in the word TAGAYTAY?

n = 8

P = 1,680

Example 2.

Eight books are to be arranged on a shelf. There are 2

Math identical books, 3 identical English books and 3

identical Physics books. How many distinct arrangement

are possible?

n = 8

= 560 arrangement

If an object may be represented by any number of times,

then the number of n different objects taken r at a time is

defined by:

nr

P =

This formula is used for permutations when repetitions are

allowed.

Example 1. In a beauty contest, 3 special prizes

are at staked to 5 contestants. If each contestants

is qualified to win all the 3 special prizes, in how

many ways can this be done?

n = 5, r= 3

nPr

35P

= 125 ways

Additional Example 4 boys and 3 girls are to be seated on a row of 7 chairs such that the boys shall occupy only the odd number chairs. Find the number of all possible ways.

Boys = 4 P 4 Girls = 3 P 3

= 4 P 4 • 3 P 3 = 24 x 6 = 144

Back to the last problem with the skiers

It can be set up as the number of permutations of 12 objects taken 3 at a time.

12P3 = 12! = 12! = (12-3)! 9!

12*11*10*9*8*7*6*5*4*3*2*1 = 9*8*7*6*5*4*3*2*1

12*11*10 = 1320

10 colleges, you want to visit all or some

How many ways can you visit

6 of them:

Permutation of 10 objects taken 6 at a time:

10P6 = 10!/(10-6)! = 10!/4! =

3,628,800/24 = 151,200

How many ways can you visit all 10 of them:

10P10 =

10!/(10-10)! =

10!/0!=

10! = ( 0! By definition = 1)

3,628,800

Permutations

A Permutation is an arrangement of items in a particular order.

Notice, ORDER MATTERS!

To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.