Mathematics Notes

1. Fundamental Principles of Counting

Addition Principle (OR)

Choose 1 item from Group A (2 items) OR Group B (3 items). Total = 2 + 3 = 5 choice.

You

Group A

Group B

Total Options: 5

Multiplication Principle (AND)

Choose 1 Shirt (2 options) AND 1 Pant (3 options). Total = 2 $\times$ 3 = 6 outfits.

Choice 1: Shirt

↓

P1P2P3

↓

P1P2P3

Total Options: 6 (2 groups of 3)

Addition Principle

If an event can occur in

m

ways and another independent event can occur in

n

ways (mutually exclusive), then either of the two events can occur in

(m + n)

ways.

Keywords: "OR", "EITHER"

Multiplication Principle

If an event can occur in

m

different ways, following which another event can occur in

n

different ways, then the total number of occurrences of the events in the given order is

(m \times n)

Keywords: "AND", "THEN"

3. Permutations (Arrangements)

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

Linear Permutation Formula

^nP_r = \dfrac{n!}{(n-r)!}

Key Scenarios

Repetition:The number of permutations of $n$ objects, where $r$ objects can be repeated, is $n^r$ .
Alike Objects:Permutations of $n$ objects where $p$ are of one kind, $q$ of another kind, etc.: $\dfrac{n!}{p! \times q! \dots}$
Circular:Number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$ .

Circular Permutation: $(n-1)!$

6 PeopleRelative Positions

Why $(n-1)!$?
In a circle, absolute positions don't matter. We must "fix" one person's position to avoid counting rotations as different.

Ways = $(6-1)! = 5! = 120$

4. Combinations (Selections)

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

The Act of Selection (Combinations)

Imagine selecting 3 balls out of 5. The order they come out doesn't change the set you have in your hand.

Selection (r=3)

Calculation

^5C_3 = \dfrac{5!}{3!2!}

= 10 ways

Combination Formula

^nC_r = \dfrac{n!}{r!(n-r)!}

ℹ️

Symmetry Property

^nC_r = ^nC_{n-r}

Pascal's Rule:

^nC_r + ^nC_{r-1} = ^{n+1}C_r

Relation between Permutations & Combinations

Relation

^nP_r = ^nC_r \times r!

Understanding the Link:

Combinations ($^nC_r$) simply SELECT elements.
Permutations ($^nP_r$) SELECT AND ARRANGE them.
Total Arrangements = (Ways to Select) $\times$ (Ways to Arrange Selection $r!$).

1. Fundamental Principles of Counting

Addition Principle (OR)

Choose 1 item from Group A (2 items) OR Group B (3 items). Total = 2 + 3 = 5 choice.

You

Group A

Group B

Total Options: 5

Multiplication Principle (AND)

Choose 1 Shirt (2 options) AND 1 Pant (3 options). Total = 2 $\times$ 3 = 6 outfits.

Choice 1: Shirt

↓

P1P2P3

↓

P1P2P3

Total Options: 6 (2 groups of 3)

Addition Principle

If an event can occur in

m

ways and another independent event can occur in

n

ways (mutually exclusive), then either of the two events can occur in

(m + n)

ways.

Keywords: "OR", "EITHER"

Multiplication Principle

If an event can occur in

m

different ways, following which another event can occur in

n

different ways, then the total number of occurrences of the events in the given order is

(m \times n)

Keywords: "AND", "THEN"

3. Permutations (Arrangements)

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

Linear Permutation Formula

^nP_r = \dfrac{n!}{(n-r)!}

Key Scenarios

Repetition:The number of permutations of $n$ objects, where $r$ objects can be repeated, is $n^r$ .
Alike Objects:Permutations of $n$ objects where $p$ are of one kind, $q$ of another kind, etc.: $\dfrac{n!}{p! \times q! \dots}$
Circular:Number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$ .

Circular Permutation: $(n-1)!$

6 PeopleRelative Positions

Why $(n-1)!$?
In a circle, absolute positions don't matter. We must "fix" one person's position to avoid counting rotations as different.

Ways = $(6-1)! = 5! = 120$

4. Combinations (Selections)

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

The Act of Selection (Combinations)

Imagine selecting 3 balls out of 5. The order they come out doesn't change the set you have in your hand.

Selection (r=3)

Calculation

^5C_3 = \dfrac{5!}{3!2!}

= 10 ways

Combination Formula

^nC_r = \dfrac{n!}{r!(n-r)!}

ℹ️

Symmetry Property

^nC_r = ^nC_{n-r}

Pascal's Rule:

^nC_r + ^nC_{r-1} = ^{n+1}C_r

Relation between Permutations & Combinations

Relation

^nP_r = ^nC_r \times r!

Understanding the Link:

Combinations ($^nC_r$) simply SELECT elements.
Permutations ($^nP_r$) SELECT AND ARRANGE them.
Total Arrangements = (Ways to Select) $\times$ (Ways to Arrange Selection $r!$).

1. Fundamental Principles of Counting

Addition Principle (OR)

Multiplication Principle (AND)

Addition Principle

Multiplication Principle

2. Factorial Notation

Definition

Special Case

Property

3. Permutations (Arrangements)

Linear Permutation Formula

Key Scenarios

Circular Permutation: $(n-1)!$

4. Combinations (Selections)

The Act of Selection (Combinations)

Combination Formula

Symmetry Property

Relation between Permutations & Combinations

Relation

5. Binomial Theorem

Binomial Expansion

General Term

📝 Practice

Concept Checks

🎲 Classic Counting: Coins, Dice, Cards

Coins

Dice

Cards

Practice Scenarios

🔥 CBSE Exam Priority Questions

Important Questions (Applied Maths)

Loading Knowledge...

1. Fundamental Principles of Counting

Addition Principle (OR)

Multiplication Principle (AND)

Addition Principle

Multiplication Principle

2. Factorial Notation

Definition

Special Case

Property

3. Permutations (Arrangements)

Linear Permutation Formula

Key Scenarios

Circular Permutation: $(n-1)!$

4. Combinations (Selections)

The Act of Selection (Combinations)

Combination Formula

Symmetry Property

Relation between Permutations & Combinations

Relation

5. Binomial Theorem

Binomial Expansion

General Term

📝 Practice

Concept Checks

🎲 Classic Counting: Coins, Dice, Cards

Coins

Dice

Cards

Practice Scenarios

🔥 CBSE Exam Priority Questions

Important Questions (Applied Maths)