Quick Revision Notes: Permutations and Combinations

Class: 11 Mathematics (MP Board)

Multiplication Principle (Fundamental Theorem of Multiplication): If an operation can be performed in $m$ different ways, and following which a second operation can be performed in $n$ different ways, then the two operations in succession can be performed in $m \times n$ ways.
Addition Principle: If an operation can be performed in $m$ different ways and another independent operation can be performed in $n$ different ways, then either of the two operations can be performed in $m + n$ ways.

Definition: The factorial of a positive integer $n$, denoted by $n!$ or $\lfloor n$, is the product of all positive integers less than or equal to $n$. $$n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1$$
Special Cases:
- $0! = 1$
- $1! = 1$
Key Property: $n! = n \times (n-1)!$

A permutation is an arrangements of a number of objects in a definite order, taken some or all at a time.

Formula for $n$ distinct objects: The number of permutations of $n$ distinct objects taken $r$ at a time ($0 \le r \le n$) is denoted by $P(n, r)$ or $^{n}P_r$: $$^{n}P_r = \frac{n!}{(n-r)!}$$
Key Deductions:
- Number of permutations of $n$ all objects at a time: $^{n}P_n = n!$
- $^{n}P_0 = 1$
- $^{n}P_1 = n$
Permutations under Restrictions:
- With Repetition: The number of permutations of $n$ distinct objects taken $r$ at a time, where repetition of objects is allowed any number of times, is $n^r$.
- With Identical Objects: The number of permutations of $n$ objects taken all at a time, where $p_1$ objects are of one kind, $p_2$ are of second kind, ..., $p_k$ are of $k^{\text{th}}$ kind, is: $$\frac{n!}{p_1! \cdot p_2! \cdot \dots \cdot p_k!}$$
- Circular Permutations:
  - Arranging $n$ distinct objects in a circle = $(n-1)!$
  - Arranging beads/garlands (clockwise and anticlockwise same): $\frac{(n-1)!}{2}$

A combination is a selection of objects where the order of selection does not matter.

Formula for $n$ distinct objects: The number of selections of $r$ objects out of $n$ objects is denoted by $C(n, r)$ or $^{n}C_r$: $$^{n}C_r = \frac{n!}{r!(n-r)!}$$
Relationship between Permutation and Combination: $$^{n}P_r = ^{n}C_r \times r!$$ $$\implies ^{n}C_r = \frac{^{n}P_r}{r!}$$
Important Properties of Combinations:
1. $^{n}C_0 = 1$ and $^{n}C_n = 1$
2. $^{n}C_1 = n$
3. Complementary Combination Rule: $^{n}C_r = ^{n}C_{n-r}$
4. Pascal's Rule: $^{n}C_r + ^{n}C_{r-1} = ^{n+1}C_r$
5. Equality Rule: If $^{n}C_x = ^{n}C_y$, then either $x = y$ or $x + y = n$.

"At least" / "At most": Use the addition/subtraction principle carefully. (e.g., At least 2 means 2 or 3 or 4...).
"And" ($\text{और}$): Indicates multiplication ($\times$).
"Or" ($\text{या}$): Indicates addition ($+$).
Formation of teams/committees: Always use Combinations ($^{n}C_r$) since order doesn't matter in a committee.
Arrangement of letters/digits: Always use Permutations ($^{n}P_r$) since order matters in words and numbers.

Permutations and Combinations