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ENM 207
Lecture 5
FACTORIAL NOTATION
The product of positive integers from 1 to n is
denoted by the special symbol n! and read
“n factorial”.
•
n!=1.2.3….(n-2).(n-1).n
•
ex: 5!=1.2.3.4.5=120
Some special factorial values
We make the following mathamatical manipulation:
Product and divide the left side of above equation by
(n-r)! and obtain n!/(n-r)!
PERMUTATIONS
Any ordered sequence of k objects taken from a
set of n distinct obfects is called a permutation
of size k of the objects.
The number of permutations of size k is obtained
from the general product rule as follows:
The first element can be chosen in n ways,
the second element can be chosen in n-1 ways,
and so on ;
PERMUTATIONS
Finally for each way of choosing the first k-1
elements, the kth element can be chosen in
n-(k-1) = n-k+1 ways, thus
The number of permutations of size k in n
distinct object is denoted by
n!
P  n.n  1n  2n  3......n  k  2n  k  1 
n  k !
k
n
COMBINATIONS
Given a set of n distinct objects any unordered
subset of size k of the objects is called a
combination.
The number of combinations of size k that can be
formed from n distinct objects will be denoted
by
   the number of combination of size k in n
n
k
COMBINATIONS
The number of combinations of size k from a
particular set is smaller than the number of
permutations because , when order is
disregarded , a number of permutations
correspond to the same combination.
C 
k
n

n
k
k
n
P
n!


k! k!n  k !
COMBINATIONS
Ex: consider the set {A,B,C,D,E} consisting of 5
elements.
We know that there are
5!/(5-3)!=60 permutations of size 3 and
5!/ 3!(5-3)!= 10 combinations of size 3
Ex: find the number of permutations of size 3
consisting of the elements of A,B,C.
3! = 3 x 2 x 1 = 6
(A,B,C) (A,C,B) (B,A,C) (B,C,A) (C,A,B) and
(C,B,A)
Ex: repititions are not permited
How many 3 digit numbers can be formed from
the six digits 2, 3, 5, 6,7 and 9?
6
5
4
numbers
i)
i)
=120
How many of these are less than 400?
2
5
4
=40
numbers
The box on the left can be filled in only two ways, by 2 or 3, since
each number must be less than 400;
The middle box can be filled in 5 ways.
The box on the right can be filled in 4 ways.
repititions are not permited
i)
How many are even?
5
4
2
Firs start filling from right side to provide condition.
The box on the right can be filled in only 2 ways
by 2 or 6, since the numbers must be even.
The box on the left can be filled 5 ways.
The box on the middle can be filled 4 ways
a) Theorem:
Let A contain n elements and let n1, n2,,,,,, nr be
positive integers with n1+ n2+ n3+ ,,,,,+ nr= n
Lets A1, A2, ...., Ar are different partitions of A
n1presents the number of elements in A1
n2 represents the number of elements in A2
and so fort nr represents the number of elements in
Ar, then there exist
n!
n1! n2 !....nr !
different ordered partitions of A.
Ex: How many distinct permutations can be formed
from all the letters of each word:
them ii) unusual iii) sociological
i)
4! = 24 , since there are 4 letters and no repitations.
ii) 7!  840 since there are 7 letters of which 3 are u
3!
iii)
12!
3!2!2!2!
since there are 12 letters of which 3
are ‘o’ , 2 are ‘c’ , 2 are ‘i’ , 2 are ‘l’
b) Theorem
b) the number of permutation of set A which
has n elements for a circle is equal (n-1)!
N people can be sit around a table in (n-1)!
different form.
Some special combinations
n!
i )C 
1
0!(n  0)!
n!
1
ii )Cn 
n
1!(n  1)!
n!
n
iii )Cn 
1
n!(n  n)!
0
n