7.2 Permutations and Combinations

Permutations and combinations are essential tools for counting complex arrangements and selections. They help us solve problems in various fields, from probability to computer science, by providing formulas to calculate different types of arrangements and selections.

These concepts build on fundamental counting principles and extend to more advanced techniques. Understanding permutations and combinations is crucial for tackling probability problems and developing problem-solving skills in discrete mathematics and beyond.

Permutations and Combinations

Fundamental Counting Principles

Top images from around the web for Fundamental Counting Principles

• 

  Counting Principles | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Permutations, Combinations, Ordered and Unordered Selections without Repetition, and Binomial ... View original

  Is this image relevant?

• 

  Chapter 6 : COUNTING PRINCIPLES (Discrete Mathematics BA202) - Engineering Mathematics 1 ... View original

  Is this image relevant?

• 

  Counting Principles | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Permutations, Combinations, Ordered and Unordered Selections without Repetition, and Binomial ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental Counting Principles

• 

  Counting Principles | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Permutations, Combinations, Ordered and Unordered Selections without Repetition, and Binomial ... View original

  Is this image relevant?

• 

  Chapter 6 : COUNTING PRINCIPLES (Discrete Mathematics BA202) - Engineering Mathematics 1 ... View original

  Is this image relevant?

• 

  Counting Principles | Algebra and Trigonometry View original

  Is this image relevant?

• 

  Permutations, Combinations, Ordered and Unordered Selections without Repetition, and Binomial ... View original

  Is this image relevant?

1 of 3

• Permutation arranges objects in a specific order
• Combination selects objects without regard to order
• Factorial notation $[n!](https://www.fiveableKeyTerm:n!)$ represents the product of all positive integers up to n
• Multiplication principle calculates the number of ways to perform a sequence of choices

Types of Permutations

• Linear permutation calculates arrangements of n distinct objects using formula $[P(n)](https://www.fiveableKeyTerm:p(n)) = n!$
• Partial permutation arranges r objects from a set of n objects, computed as $P(n,r) = \frac{n!}{(n-r)!}$
• Circular permutation organizes n objects around a circle, given by $[C(n)](https://www.fiveableKeyTerm:c(n)) = (n-1)!$
• Derangement counts permutations where no element appears in its original position, calculated using $[D_n](https://www.fiveableKeyTerm:d_n) = n! \sum_{k=0}^n \frac{(-1)^k}{k!}$

Combination Principles

• Combination formula selects r objects from n objects without order: $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
• Pascal's Triangle illustrates combinations and binomial coefficients
• Combination properties include symmetry: $\binom{n}{r} = \binom{n}{n-r}$
• Vandermonde's Identity relates combinations: $\binom{m+n}{r} = \sum_{k=0}^r \binom{m}{k}\binom{n}{r-k}$

Permutations and Combinations with Repetition

Permutations with Repetition

• Permutation with repetition allows objects to be used multiple times
• Formula for n objects into r positions: $n^r$
• Multinomial coefficient calculates permutations with multiple types of repeated objects: $\frac{n!}{n_1!n_2!...n_k!}$
• Applications include generating passwords and license plate numbers

Combinations with Repetition

• Combination with repetition selects objects with replacement, allowing repeated selections
• Formula for selecting r objects from n types: $\binom{n+r-1}{r} = \binom{n+r-1}{n-1}$
• Stars and bars method visualizes combinations with repetition
• Used in problems involving distributing identical objects into distinct containers

Advanced Counting Techniques

Stirling Numbers and Their Applications

• Stirling numbers of the first kind count permutations with specific cycle structures
• Stirling numbers of the second kind partition a set into non-empty subsets
• Recursive formulas define both types of Stirling numbers
• Bell numbers sum Stirling numbers of the second kind to count total partitions of a set
• Applications include analyzing data structures and solving combinatorial problems in computer science