Quants: Permutation and Combination

Study Notes

Quants: Permutation and Combination

Definitions

Permutation: An arrangement of objects in a specific order. Order matters.
Combination: A selection of objects without regard to the order. Order does not matter.

Permutation

Formula:
- ( P(n, r) = \frac{n!}{(n - r)!} )
- Where ( n ) = total number of items, ( r ) = number of items to arrange, ( ! ) denotes factorial.
Types:
- Distinct Permutations: All items are different.
- Repeating Permutations: Some items are alike.
  - Formula: ( P(n; n_1, n_2, \ldots, n_k) = \frac{n!}{n_1! \cdot n_2! \cdots n_k!} )
Examples:
- Arranging 3 books from a shelf of 5: ( P(5, 3) = \frac{5!}{(5 - 3)!} = 60 )
- Arranging the letters of “BALLOON”: ( P(7; 1, 1, 2, 2, 1) = \frac{7!}{1!1!2!2!1!} = 420 )

Combination

Formula:
- ( C(n, r) = \frac{n!}{r!(n - r)!} )
- Where ( n ) = total items, ( r ) = items chosen.
Types:
- Distinct Combinations: All items are different.
- Repeating Combinations: Some items can be chosen more than once.
  - Formula: ( C(n + r - 1, r) = \frac{(n + r - 1)!}{r!(n - 1)!} )
Examples:
- Choosing 3 fruits from 5: ( C(5, 3) = \frac{5!}{3!2!} = 10 )
- Choosing 2 toppings from 3 available types (with repetition allowed): ( C(3 + 2 - 1, 2) = C(4, 2) = 6 )

Key Differences

Order:
- Permutation: Order matters.
- Combination: Order does not matter.
Usage:
- Permutations are used in scenarios like race placements or password formation.
- Combinations are used in scenarios like lottery selections or committee formations.

Tips for Solving Problems

Identify whether order matters (use permutation) or does not matter (use combination).
Use factorial calculations to compute the number of arrangements or selections.
Break complex problems into smaller parts to simplify calculations.

Definitions

Permutation: Arrangements where order is significant.
Combination: Selections where order is irrelevant.

Permutation

Formula: ( P(n, r) = \frac{n!}{(n - r)!} )
- ( n ): Total items, ( r ): Items to arrange, ( ! ): Factorial notation.
Types:
- Distinct Permutations: All items are unique.
- Repeating Permutations: Includes items that appear more than once.
- Repeating Permutation Formula: ( P(n; n_1, n_2, \ldots, n_k) = \frac{n!}{n_1!\cdots n_k!} )
Examples:
- Arranging 3 books from 5 yields ( P(5, 3) = 60 ).
- Arranging letters in “BALLOON” leads to ( P(7; 1, 1, 2, 2, 1) = 420 ).

Combination

Formula: ( C(n, r) = \frac{n!}{r!(n - r)!} )
- ( n ): Total items, ( r ): Items selected.
Types:
- Distinct Combinations: All selections are unique.
- Repeating Combinations: Selections allow for duplicates.
- Repeating Combination Formula: ( C(n + r - 1, r) = \frac{(n + r - 1)!}{r!(n - 1)!} )
Examples:
- Choosing 3 fruits from 5 results in ( C(5, 3) = 10 ).
- Selecting 2 toppings from 3 (with repeats) gives ( C(4, 2) = 6 ).

Key Differences

Order:
- Permutation: Order is crucial.
- Combination: Order is unimportant.
Usage:
- Permutations: Suitable for situations like race rankings or password setups.
- Combinations: Appropriate for lottery picks or committee selections.

Tips for Solving Problems

Assess if arrangement or selection matters to determine whether to use permutations or combinations.
Apply factorial calculations for determining counts of arrangements or selections.
Simplify complex problems by breaking them down into manageable components.

Description

Test your understanding of permutations and combinations with this quiz. Dive into the definitions, formulas, and types of arrangements and selections that make up these fundamental concepts in combinatorics. Perfect for students looking to reinforce their knowledge in mathematics.