Two fundamental principles of counting:
Addition Rule :
If an experiment can be performed in n ways, & another
experiment can be performed in m ways then either of the two
experiments can be performed in n+m ways.
Multiplication Rule :
If a work can be done in n ways, another work can be done in m
ways, then both of the operations can be performed in n x m
ways.
Example
Suppose there are 3 doors in a room,
2 on one side and 1 on other side. A
man want to go out from the room.
Obviously he has ‘3’ options for it. He
can come out by door ‘A’ or door ‘B’
or door ’C’.
Suppose a man wants to cross-out a
room, which has 2 doors on one side
and 1 door on other site. He has 2 x 1
= 2 ways for it.
Factorial !
Factorial n: the product of first n natural numbers is
denoted by n!
n! = n(n-1) (n-2) ………………..3.2.1.
Permutations
• Permutation of a set of objects is an arrangement of
those objects into a particular order.
• That means in permutation, the order matters.
The permutation of n objects out of n (without repetition) is n!
The 6 permutations of 3 balls
Permutation of r objects from n objects
1. With repetition :
nr
e.g. Choosing a 3-digit code for a lock from (0,1,2,.....,9)
There are 103 = 1000 choices
Combination locks
(wrong name)
Permutation of r objects from n objects
2. Without repetition :
n!
(n  r )!
e.g. Choosing a 3-digit code for a lock from (0,1,2,.....,9)
There are
10!
 720
(10  3)!
Combinations
• It is a way of selecting r objects from n objects,
where the order does not matter.
1. Combination without repetition
n
C
r
n!
r !(n  r )!
4
e.g. Choosing 3 letters from A, B, C and D:
C
3
4!
4
3!(4  3)!
(this can also be calculated by using Pascal's triangle)
1. Combination with repetition
It is hardest to calculate
(n  r  1)!

r !(n  1)!
rn
Conclusion
(selecting r objects from a group of n objects)
• Repetition allowed, order matters:
• No repetition, order matters:
n
r
n!
(n  r )!
• Repetition allowed, order doesn’t matter:
• No repetition, order doesn’t matter:
(n  r  1)!
r !(n  1)!
n!
r !(n  r )!
Nine players take part in a competition. In how
many ways the first three places can be taken?
504
In how many ways can a committee of 5
be chosen from 10 people?
252
Jones is the Chairman of a committee. In how many
ways can a committee of 5 be chosen from 10
people given that Jones must be one of them?
126
A password consists of four different letters of the
alphabet. How many different possible passwords
are there?
358,800
A password consists of two letters of the alphabet
followed by three digits chosen from 0 to 9. Repeats
are allowed. How many different possible passwords
are there?
676,000
Circular permutation
• If clockwise and anti-clockwise orders are regarded different,
then the number of permutations is (n  1)!
• If they are regarded the same (flipping the circle), then the
number of permutations is just 1 for n  1, 2
and 1 (n  1)! for n  3
2
Examples on restricted- circular permutations
A) In how many ways can 4 married couples seat themselves
around a circular table if spouses sit opposite each other?
B) Mom and dad and their 6 children (3 boys and 3 girls) are to
be seated at a table. How many ways can this be done if mom
and dad sit together and the males and females alternate?
a)
b)
48
72