Combinatorial Analysis
Definition :
The combinatorial analysis is the science that studies the formation of sets in all their forms, whether they are ordered or unordered.
Fundamental Counting Principle
Suppose that two events occur in order. If the first can occur in m ways and the second in n ways (after the first has occurred), then the two events can occur in order in m×n ways.
Definition of factorial notation
Definition :
The product of the first n natural numbers is denoted by n! and is called n factorial:

Note :
Zero factorial is defined as follows:
An alternative is to define n! recursively on the non-negative integers

As n increases, n! increases very rapidly.
For any fixed number a, for all n sufficiently large, on the other hand
for all n.
Example :
Calculate the following:

Note :

Permutations
A permutation[1] is: an ordering of a number of distinct items in a line is denoted by (P[2]). Sometimes even though we have a large number of distinct items, we want to single out a smaller number and arrange those into a line; this is also a sort of permutation.
Definition :
A permutation of n distinct objects is an arrangement of those objects into an ordered line. If 1≤k≤n (and k is a natural number) then an k-permutation of n objects is an arrangement of k of the n objects into an ordered line .
Example :
There are 7 horses in a race
1/ In how many different orders can the horses finish?
We choose 7 horses from 7, so we use the definition of permutation as follow:

2/ In how many ways we choose the first,second, and third?
Here 3-permutations of 7 horses, there are 7 ways to choose the
6 ways to choose the second and 5 ways to choose the third.

We can use the same reasoning to determine a general formula for the number of k-permutations of n objects:
Theorem
Example :
How many arrangements of the letters of the word REMAND are possible?

How many arrangements of the letters of the word PARRAMATTA are possible?
In this word there is some repeated letter, so we are going to use the following rule:

A person wanted to create a password for his email. What is the number of words that can be created that consist of
a/ 3 numbers with repetition
We note that we are faced with the following conditions
1/ The part of the whole
2/ Order is important
3/ Repetition is allowed
So we use k-permutations (arrangement) with repitition

b/ 5 numbers without repetition
We note that we are faced with the following conditions
1/ The part of the whole
2/ Order is important
3/ Repetition is not allowed
So we use k-permutations (arrangement) without repitition

Properties

Example :
Calculate the following:

Combinations
Sometimes the order in which individuals are chosen doesn't matter; all that matters is whether or not they were chosen.
Definition :
Let n be a positive natural number, and 0≤k≤n. Assume that we have n distinct objects. An k-combination[5] of the n objects is a subset consisting of k of the objects. So a combination involves choosing items from a finite population in which every item is uniquely identified, but the order in which the choices are made is unimportant, this one is denoted by (C[4]).
We can use the same reasoning to determine a general formula for the number of k-combinations of n objects:
Theorem
The number of k-combinations of n objects is

Notation
We use to denote the number of k-combinations of n objects, so:

Note :
We read as " n choose k" so n choose k is
,Notice that when k=n, we have

coinciding with our earlier observation that there is only one way in which all of the n objects can be chosen. Similarly,

there is exactly one way of choosing none of the n objects.
Properties
1/

2/ For n,p∈ℕ and 0≤p≤n we have:

3/ For n,p∈ℕ and 1≤p≤n we have:

Example :
We want to form a student committee of 5 students out of 10 students at the first year level and 15 students at the second year level.
a/ to find how many ways we can form this committee, we note that we are faced with the following conditions
The part of the whole
∙Order is not important
Repetition is not allowed
So we use k-combinations without repitition:

b/ Calculating the number of appropriate cases so that the committee includes two students from the first year level and three students from the second year level
we note that we are faced with the following conditions
The part of the whole
Order is not important
Repetition is not allowed
So we use k-combinations without repitition

Example :
Calculate the following:

Binomial Theorem
Theorem
For any a and b, and any natural number n, we have

One special case of this is that

Example :
Use the Binomial Theorem to evaluate the following:
