1. PERMUTATIONS and COMBINATIONS
QS 026 CHAPTER 9
PERMUTATIONS and COMBINATIONS
PERLIS MATRICULATION COLLEGE

2. Permutations of a set of objects
A permutation of a set of objects is any arrangement of those objects in a definite order.
(order is important).
For example, if A={a,b,c,d} , then
ab two-element permutation of A,
acd three-element permutation of A,
adcb four-element permutation of A.

3. The order in which objects are arranged is important.
For example,
ab and ba are considered different two-
element permutations
abc and cba are distinct three-element
permutations, and
abcd and cbad are different four-element
permutations.

4. For another example the six permutations of ABC are the six different arrangements of ABC. These are
ABC ACB BAC
BCA CAB CBA

5. If there are 4 ways from Johor to Penang and 2 ways from Penang to Pulau Langkawi, how many ways can we go for a journey from Johor to Pulau Langkawi through Penang.

6. Johor Train Penang Ferry P.Langkawi
Bus
Taxi Flight
Johor Train Penang Ferry P.Langkawi
Van

7. The number of permutations can be calculated using the multiplication principle.
If there are m ways for an event to occur and n ways for another event to occur, then there are m x n ways for the two events to occur.

8. Example
A coin and a dice are tossed together. How many different outcomes are possible ?
Solution
The coin has two possible outcomes (head, H and Tail, T) and the dice has 6 possible outcomes.
The number of different possible outcomes is
___ x ___ = ____ 2 6 12
The possible outcomes are
(H,1)
(H,2)
(H,3)
(H,4)
(H,5)
(H,6)
(T,1)
(T,2)
(T,3)
(T,4)
(T,5)
(T,6)

9. Die 1 2 3 4 5 6
Coin
Head (H)
(H,1)
(H ,2)
(H, 3)
(H, 4)
(H, 5)
(H, 6)
Tail (T)
(T,1)
(T,2)
(T, 3)
(T, 4)
(T, 5)
(T, 6)

10. Example
A shop stocks T-shirts in four sizes : small, medium and large. They are available in four colours; black , red , yellow and green. If the sizes are denoted by S, M and L and the colours are denoted by B, R, Y and G make a list of all the different labels needed to distinguish the T-shirts and find the number of different labels.
Solution
SB SR SY SG MB MR
MY MG LB LR LY LG
The number of different labels is 3 x 4 = 12

11. Permutations of n objects
We will now consider the method for finding a number of permutations on the letters A, B and C using the multiplication principle. How many arrangements of the letters A, B and C are there?
Let us consider the number of ways of arranging n letter.
If we have 1 letter, there is just one arrangement. Example : A

12. If we have 2 letters, there are two different arrangements
If we have 2 letters, there are two different arrangements. Example : AB and BA
If we have 3 letters, the different arrangements are :
there are three ways of
choosing the first letter.

13. When the first letter has been chosen, there are two letters from which to choose the second; and the possible ways of choosing the first two letters are:
there are two ways of choosing
the second letter

14. i.e. for each of the three ways of choosing the first letter, there are two ways of choosing the second letter.
Hence there are 3 x 2 ways of choosing the first two letters.
Having chosen the first two letters, there is only one choice for the third letter, i.e. for each of the 3 x 2 ways of choosing the first two letters, there is only one possibility for the third letter.
Hence there are 3 x 2 x 1 ways of arranging the three letters A, B and C.
Note : if repetition are allowed, we can choose from all 3 digits for each digit of the number. A digit can be used more than once.

15. How many different ways of arranging 3 digit
numbers from digits 5 and 6 ? 5 6
there will be 8 different ways, which is found from

16. How many different ways do you think there
are of arranging 4 letters?
You should able to see, there will be 24 different
ways, which is found from 4 x 3 x 2 x 2 x 1.
If there are 500 different objects, the number of
ways would be 500 x 499 x 498 x … x 3 x 2 x 1.
This is tedious to write, so we use the notation
500! ( factorial 500 ).

17. Example
List the set of all permutations of the symbols P, Q and R when they are taken 3 at a time
Solution
PQR,
PRQ,
QPR,
QRP,
RPQ,
RQP

18. Notes : n! means the products of all the
In general,
Number of permutations of n different objects taken all at a time without repetition n
= n x (n – 1) x(n – 2) x … x 2 x 1
= n! P n n P
= n! n
Notes : n! means the products of all the
integers from 1 to n inclusive and is called ‘n factorial’.

19. How many three-digit numbers can be made from the integers 2, 3, 4 ?
Example
How many three-digit numbers can be made from the integers 2, 3, 4 ?
Solution
n = 3 n
= 3 ! 3 x 2 x 1 = 6 3 = n
The number of arrangements is 6.

20. Example
In how many ways can ten instructors be assigned to ten sections of a course in mathematics?
Solution
Substituting n = 10 we get
= 10 ! = 3,628,800 ways =

21. Example
Three people, Aishah, Badrul and Daniel must be scheduled for job interviews. In how many different orders can this be done?
Solution
n = 3
So there are 3! = 6 possible orders for the interviews.

22. Example
How many different numbers can be formed
from the digits 5, 6, 7 and 8
i)   if no repetitions are allowed
n = 4
npn = 4p4 = 4! = 24
ii)  if the first digit must be 7.
1p1 X 3p3 = 1! X 3! = 6