Permutation, Combination & Probability

Factorial Notation:
The factorial notation was introduced by Christian Kramp (1760 - 1826) in 1808.

n! = n(n - 1)(n - 2)(n - 3).......3.2.1

For example,

1! = 1

2! = 2.1 = 2

3! = 3.2.1 = 6

4! = 4.3.2.1 = 24

5! = 5.4.3.2.1 = 120

6! = 6.5.4.3.2.1 = 720 

Important Note:

Thus for a positive integer n, we define n factorial as:

n! = n(n - 1)!           where 0! = 1

Permutation:
An ordering (arrangement) of n objects is called a permutation of the objects.
A permutation of n different objects is an ordering (arrangement) of the objects such that one object is first, one is second, one is third and so on.
According to the Fundamental Principle Of Counting:

1. Three books can be arranged in a row taken all at a time = 3.2.1 = 3! ways.
2. Number of ways of writing the letters of the WORD taken all at a time = 4.3.2.1 = 4! 

Each arrangement is called a permutation. Now we have the following definition.
"A Permutation of n different objects taken r (≤ n) at a time is an arrangement of the r objects. Generally it is denoted by ⁿP_r or P(n , r)".

ⁿP_{r = n! / (n - r)!}

Combinations:

While counting the number of possible permutation of a set of objects, the order is important. But there are situations where order is immaterial. For example;

1. ABC, ACB, BAC, BCA, CAB, CBA are the six names of the triangle whose vertices are A, B and C. We notice that inspite of the different arrangements of the vertices of the triangle, they represent one and the same triangle.
2. The 11 players of a cricket team can be arranged in 11! ways, but they are players of the same single team.

So, we are interested in the membership of the committee (group) and not in the way the members are listed (arranged).

Therefore, a combination of n different objects taken r at a time is a set of r objects.

The number of combinations of n different objects taken r at a time is denoted by ⁿC_r or C(n , r) and is given by:

ⁿC_r = n! / r! (n - r)!

Probability:

Probability is the numerical evaluation of a chance that a particular event would occur.

P(E) = n/m = n(E) / n(S)

P(E) = no. of ways in which event occurs / no. of the elements of the sample space