Permutation and Combination is a fundamental topic in combinatorics, widely used in JEE Main and Advanced. It deals with counting arrangements and selections without explicitly listing them.
Basic Principle of Counting:
If a task can be performed in \( m \) ways and another independent task in \( n \) ways, then both together can be performed in:
\[
m \times n
\]
This is known as the Fundamental Principle of Counting.
Factorial Notation:
Factorial is defined as:
\[
n! = n \times (n-1) \times (n-2) \times \cdots \times 1
\]
Special case:
\[
0! = 1
\]
Permutations:
A permutation is an arrangement where order matters.
Number of permutations:
\[
{}^nP_r = \frac{n!}{(n-r)!}
\]
Example:
\[
{}^5P_3 = \frac{5!}{2!} = 60
\]
Permutations with Repetition:
If repetition is allowed:
\[
n^r
\]
Circular Permutations:
Number of arrangements in a circle:
\[
(n – 1)!
\]
Combinations:
A combination is a selection where order does not matter.
Number of combinations:
\[
{}^nC_r = \frac{n!}{r!(n-r)!}
\]
Example:
\[
{}^5C_3 = \frac{5!}{3!2!} = 10
\]
Relation Between Permutation and Combination:
\[
{}^nP_r = {}^nC_r \times r!
\]
Important Identities:
1. Symmetry property:
\[
{}^nC_r = {}^nC_{n-r}
\]
2. Pascal’s Identity:
\[
{}^nC_r = {}^{n-1}C_r + {}^{n-1}C_{r-1}
\]
Combinations with Repetition:
\[
{}^{n+r-1}C_r
\]