Ch 06 - Permutations and Combinations11th Mathematics
Fundamental Principle of Counting (Multiplication Rule)
What is the Fundamental Principle of Counting (Multiplication Rule) and when is it strictly applied?
What is the Fundamental Principle of Counting (Multiplication Rule) and when is it strictly applied?
Core Concept
If an event can occur in different ways, following which another independent event can occur in different ways, then the total number of occurrences of the events in the given sequence is .
Boundary Conditions / Domain
- The events must be sequential and independent (i.e., 'AND' condition).
- The choice made in the first event must not restrict the available number of ways in the second event unless explicitly accounted for.
Prism Memory Hook
AND means MULTIPLY.
Addition Rule vs Multiplication Rule
In combinatorics, what is the strict mathematical distinction between applying the Addition Rule versus the Multiplication Rule?
In combinatorics, what is the strict mathematical distinction between applying the Addition Rule versus the Multiplication Rule?
Multiplication Rule
Used when tasks are sequential and all must be performed to complete the operation (e.g., Task A AND Task B). Total ways: .
Addition Rule
Used when tasks are mutually exclusive and performing any one task completes the operation (e.g., Task A OR Task B). Total ways: .
Boundary Conditions / Domain
- For the Addition Rule, the sets of outcomes must be disjoint (no intersection). If they overlap, you must subtract the intersection (Principle of Inclusion-Exclusion).
Prism Memory Hook
AND = Multiply, OR = Add.
Factorial Notation Definition (n!)
Factorial Notation Definition (n!)
Mathematical definition and formula for n factorial (n!).
Derivation
Note the critical recursive property used for simplification:
Component Anatomy
- : Represents the continued product of the first natural numbers.
Boundary Conditions / Domain
- MUST be a non-negative integer (). Factorials are not defined for negative integers or fractions in standard introductory combinatorics.
Prism Memory Hook
A premium mnemonic for this topic is available in the Prism App.
Also in this chapter
43 more cards to study in Prism
- 4Mathematical Value of 0!
- 5Mathematical Definition of a Permutation
- 6Mathematical Definition of a Combination
- 7Permutations of Distinct Objects without Repetition
- 8Permutations with Repetition Allowed
- 9Permutations Involving Identical Objects (One Kind)
- 10Permutations Involving Identical Objects (Multiple Kinds)
- 11Circular Permutations of Distinct Objects
- 12Circular Permutations with Reversible Orientation (Necklaces)
- 13Combinations Formula (Selection of Distinct Objects)
- 14Algebraic Relationship Between Permutations and Combinations
- 15Complementary Combination Property
- 16Combination Equality Theorem
- 17Pascal's Rule for Combinations
- 18Ratio of Consecutive Combinations
- 19SOP: String / Tie Method (Objects Always Together)
- 20SOP: Gap Method (No Two Objects Together)
- 21SOP: ALL Restricted Objects Are Not Together (Complementary)
- 22SOP: Arrangements with a Specific Object ALWAYS Included
- 23SOP: Arrangements with a Specific Object NEVER Included
- 24SOP: Dictionary Rank of a Word (Distinct Letters)
- 25SOP: The Sliding Window Trick (Fixed Gap Between Objects)
- 26SOP: Sum of All Formed n-Digit Numbers
- 27Selection of 'One or More' (Distinct vs Identical)
- 28SOP: Selection from Multiple Identical Subsets
- 29The 'Select THEN Arrange' Paradigm
- 30Combinatorics of Handshakes and Chords
- 31Maximum Intersections of n Straight Lines
- 32Number of Diagonals in an n-Sided Polygon
- 33Collinear Points: Forming Straight Lines
- 34Collinear Points: Forming Triangles
- 35Total Squares on an n x n Grid
- 36Division of Distinct Objects into Unequal Groups
- 37Division of Distinct Objects into Equal Unordered Groups
- 38Distinction: Group Division vs Group Distribution
- 39Unrestricted Distribution of Distinct Objects
- 40Restricted Distribution of Distinct Objects (Max 1 per Recipient)
- 41Stars and Bars: Distribution of Identical Objects (Blanks Allowed)
- 42Stars and Bars: Distribution of Identical Objects (Blanks NOT Allowed)
- 43Derangement Formula D(n)
- 44Partial Derangement
- 45The 'All or None' Group Selection Trap
- 46Simplification: Product of First n Even Natural Numbers
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