**Fundamental Counting Principle:** If one work can be done by "a" different styles and a second work can be done by "b" different styles then the total syles of doing both works will be \((a \times b)\).

**Factorial:** The factorial of an integer is the product of all less than or equal positive integers. It is denoted by \(n!\), where n is any positive integer. The factorial of a negative integer is not possible.$$ n! = n \ (n - 1) \ (n - 2)......3 \times 2 \times 1 $$ $$ OR $$ $$ n! = 1 \times 2 \times 3.........(n - 1) \times n $$ $$ n! = (n - 1)! \times n $$

**Example:** \(4! = 4 \times 3 \times 2 \times 1 = 24\)

**Example:** \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\) \(= 720\)

**Note:** Remember, \(0! = 1\), \(1! = 1\), \(nP_0 = 1\), \(nP_n = n!\).

**Permutation:** The selection of objects in an ordered manner is called a permutation.

**Example:** Three pens A, B, and C can be written in six different orders ABC, ACB, BAC, BCA, CAB, CBA Hence here the total number of permutation are 6.

**Notation of Permutation:** If there are "n" distinct objects and we select "r" objects together then the number of permutations is denoted as \(nP_r\) or P(n, r). $$ nP_r = n \ (n - 1) \ (n - 2)......r $$ $$ nP_r = \frac{n!}{(n - r)!} $$ Here (r < n).

Where, n = Total number of objects.

P = Permutation.

r = The number of selected objects at a time from the total number of distinct objects.

**Important Formulae:**

**(i):** n = total number of objects.

r = Number of selected objects at a time from the total number of distict objects. $$ nP_r = \frac{n!}{(n - r)!} $$

**(ii):** n = total number of objects.

p, q, r = the number of different selected objects at a time from the total number of distinct objects. $$ P(p, q, r) = \frac{n!}{p! q! r!} $$

**Example:** Find the value of \(10P_2\)?

**Solution:** $$ nP_r = \frac{n!}{(n - r)!} $$ $$ 10P_2 = \frac{10!}{(10 - 2)!} $$ $$ = \frac{10!}{8!} $$ $$ = \frac{10 \times 9 \times 8!}{8!} $$ $$ 10P_2 = 90 $$

Arithmetic Progression
Geometric Progression
Harmonic Progression
Partial Fractions
Theory of Equations
Permutation and Combination
Binomial Theorem
Matrices