2. 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.
3. 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
4. In general, the # of permutations of n
objects is taken n at a time is:
5. Try…
1. 8 P 8 = 8! =
2. 5 P 5 = 5! =
3. 4 P 4 = 4! =
6. Example 1.
In how many ways can a boy arrange his 5 different
toys in a row?
n = 5
5P5 = 5!
= 120
7. 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
8. 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
9. The number of Permutation of n different objects taken r at a
time is denoted and defined, as follows:
11. 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
12. 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(
!30
330
27!
30!
p
13. 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*15
12!
15!
)!315(
!15
315
14. 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:
15. 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
16. 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
17. 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:
18. Example 1
How many permutations are there in the
word TAGAYTAY?n = 8
P = 1,680
19. 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
20. 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:
n
r
P =
This formula is used for permutations when repetitions are
allowed.
21. 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
nP
r
3
5P
= 125 ways
22. 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
23. 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
24. 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
25. How many ways can you visit all 10
of them:
10P10 =
10!/(10-10)! =
10!/0!=
10! = ( 0! By definition = 1)
3,628,800
26. 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.
