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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 :

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Enm 207

ENM 207

Lecture 5


Factorial notation
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
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
PERMUTATIONS

Any orderedsequence 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 ;


Permutations1
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 kin n distinct object is denotedby


Combinations
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


Combinations1
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.


Combinations2
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!=3x2x1=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

2

5

4

=40 numbers

6

5

4

=120 numbers

Ex: repititions are not permited

How many 3 digit numbers can be formed from the six digits 2,3,5,6,7 and 9?

i)

i)How many of these are less than 400?

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

5

4

2

repititions are not permited

i)                    How many are even?

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
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

different ordered partitions of A.


E x how many distinct permutations can be formed from all the letters of each word
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.

  • since there are 7 letters of which 3 areu

  • since there are 12 letters of which 3

    are ‘o’ , 2 are ‘c’ , 2 are ‘i’ , 2 are ‘l’


B theorem
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.



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