Glossary

7.2 Permutations

2 min read•june 18, 2024

are all about things in different orders. They help us count how many ways we can line up objects or pick items in a specific sequence. This concept is super useful for solving real-world problems and understanding probabilities.

We use notation and special formulas to calculate permutations. These tools let us figure out things like how many ways to arrange letters in a word or pick winners in a contest. It's a key part of discrete math and theory.

Permutations

Factorial formula for permutations

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  • Calculates the number of ways to arrange nn distinct objects in a specific order
    • Multiplies all positive integers from 1 up to nn (5 objects: 5×4×3×2×1=1205 \times 4 \times 3 \times 2 \times 1 = 120 arrangements)
  • Accounts for repeated objects by dividing [n!](https://www.fiveableKeyTerm:n!)[n!](https://www.fiveableKeyTerm:n!) by the factorial of each repeated object's count
    • Arranging "MISSISSIPPI" (44 I's, 44 S's, 22 P's, 11 M): 11!4!×4!×2!×1!=34,650\frac{11!}{4! \times 4! \times 2! \times 1!} = 34,650 unique permutations
    • Eliminates duplicate arrangements caused by identical objects (e.g., swapping positions of two I's)
  • Utilizes principles to define the elements being arranged

Permutation notation for subsets

  • Denotes the number of ways to select and arrange rr objects from a set of nn distinct objects
    • Notated as [P(n,r)](https://www.fiveableKeyTerm:P(n,r))[P(n, r)](https://www.fiveableKeyTerm:P(n,_r)) or nPr_{n}P_{r}, calculated as n!(n−r)!\frac{n!}{(n-r)!}
    • Choosing 33 people from a group of 77 to stand in a line: P(7,3)=7!(7−3)!=210P(7, 3) = \frac{7!}{(7-3)!} = 210 possible arrangements
  • Applies when order matters and repetition is not allowed
    • Arranging books on a shelf, selecting contest winners (first, second, third place)
  • Contrasts with , where order doesn't matter

Permutations in probability problems

  1. Determine the total number of possible outcomes (nn) and desired outcomes (rr)
  2. Calculate the number of ways the desired outcome can occur using the appropriate formula
  3. Divide the number of desired outcomes by the total outcomes to find the probability
  • Winning a lottery by matching 66 numbers drawn from 4949 in the correct order:
    • Total outcomes: P(49,6)=49!(49−6)!=10,068,347,520P(49, 6) = \frac{49!}{(49-6)!} = 10,068,347,520
    • Desired outcome: 11 (only one way to match all 66 numbers in the correct order)
    • Probability: 110,068,347,520≈9.93×10−11\frac{1}{10,068,347,520} \approx 9.93 \times 10^{-11} (about 11 in 1010 billion)
  • Permutations help calculate probabilities for ordered outcomes (e.g., horse race results, card hands)
  • : Permutations are a fundamental counting method in
  • Probability: Permutations are essential for calculating probabilities of ordered events
  • Combinations: Another counting technique used when order doesn't matter, complementing permutations

Key Terms to Review (22)

Arranging: Arranging refers to the process of organizing or putting items in a specific order or sequence. In mathematics, particularly in permutations, arranging is about determining the different ways in which a set of items can be ordered, where the order of selection matters. This concept is crucial for solving problems involving combinations and sequences, as it allows us to count and analyze different arrangements effectively.
Bijective proof: A bijective proof is a mathematical argument that demonstrates the equivalence of two sets by establishing a one-to-one correspondence between them. This type of proof relies on the concept of bijections, which are functions that pair each element of one set with exactly one element of another set, ensuring that every element is accounted for and mapped uniquely. Bijective proofs are particularly useful in combinatorics, including when dealing with permutations.
Circular permutation: Circular permutation refers to the arrangement of objects in a circle where the order of arrangement matters, but rotations of the same arrangement are considered identical. This concept is particularly relevant when analyzing arrangements that form a loop, as it simplifies counting by recognizing that rotating the entire arrangement does not create a new unique arrangement.
Combination: A combination is a selection of items from a larger set where the order of selection does not matter. Understanding combinations helps in various scenarios such as calculating probabilities, forming groups, and organizing outcomes where the sequence is irrelevant, linking directly to concepts like counting rules, permutations, and probability calculations.
Combinations: Combinations refer to the selection of items from a larger set where order does not matter. They are used to determine how many ways a subset of items can be chosen from the entire set without regard to the sequence of selection.
Combinatorial proof: A combinatorial proof is a method of proving mathematical identities or statements by counting the same set in two different ways. This technique uses combinatorial reasoning, often relying on the concept of counting the number of ways to arrange or select items, which can be particularly useful in the context of permutations. By establishing a clear relationship between two counting methods, one can demonstrate the truth of an equation or identity without resorting to algebraic manipulation.
Counting Techniques: Counting techniques are systematic methods used to count or enumerate objects, outcomes, or arrangements in mathematics. These techniques include strategies such as permutations and combinations, which help in determining the total number of ways to arrange or select items from a set. Understanding counting techniques is crucial for solving problems involving probability, combinatorics, and decision-making.
Discrete Mathematics: Discrete mathematics is a branch of mathematics that deals with countable, distinct objects and structures. It involves concepts such as integers, graphs, and logical statements, making it essential for computer science, information theory, and combinatorial optimization. This area of mathematics emphasizes finite structures rather than continuous ones, leading to applications in algorithm design, cryptography, and network theory.
Empirical probability: Empirical probability is the probability of an event determined by conducting experiments or observing real-life occurrences. It is calculated as the ratio of the number of favorable outcomes to the total number of trials.
Factorial: A factorial, denoted by the symbol 'n!', is a mathematical operation that multiplies a whole number by all of the positive whole numbers less than it. Factorials are essential in counting arrangements and selections, making them pivotal in understanding permutations and combinations. The concept extends to the multiplication rule for counting as it helps in calculating the total number of ways to arrange or select items.
Fundamental counting principle: The fundamental counting principle states that if one event can occur in 'm' ways and a second event can occur independently in 'n' ways, then the total number of ways that both events can occur is the product of the two numbers, or 'm × n'. This principle is foundational in combinatorics and helps simplify the process of counting outcomes in various scenarios.
Linear permutation: A linear permutation is an arrangement of a set of items in a specific order, where the sequence matters. This concept is fundamental in combinatorics and helps in understanding how many ways items can be ordered without repetition. Linear permutations can be calculated using factorial notation, which emphasizes the significance of each position in the arrangement.
Multiplication principle: The multiplication principle is a fundamental rule used to determine the number of possible outcomes in a sequence of events. It states that if one event can occur in 'm' ways and a second event can occur independently in 'n' ways, then the two events together can occur in m * n ways.
Multiplication Principle: The multiplication principle states that if there are multiple independent choices to be made, the total number of possible outcomes is found by multiplying the number of options for each choice. This principle is fundamental in counting scenarios where the arrangement or selection of items occurs in sequences, particularly in contexts involving arrangements and selections of groups.
N!: The notation 'n!' represents the factorial of a non-negative integer n, defined as the product of all positive integers less than or equal to n. This concept is crucial in combinatorial mathematics, as it is used to calculate the number of ways to arrange or select items from a set, forming the foundation for understanding arrangements and selections in various contexts.
NPr: nPr represents the number of permutations of n distinct objects taken r at a time. It quantifies how many different ways you can arrange r objects selected from a larger group of n unique items. Understanding nPr is essential for solving problems that involve arrangements or orderings, as it helps determine the count of possible configurations when the order matters.
Ordering: Ordering refers to the arrangement or sequence of elements based on a certain criterion, which can be numerical, alphabetical, or based on other properties. This concept is fundamental in understanding how different entities relate to one another and is crucial for organizing information clearly and effectively. It allows for the comparison of values and the establishment of hierarchy within a set.
P(n, r): P(n, r) represents the number of permutations of 'n' distinct objects taken 'r' at a time. It is a mathematical expression used to calculate how many different ways we can arrange 'r' items from a set of 'n' items, highlighting the importance of order in these arrangements.
Permutation: A permutation is an arrangement of objects in a specific order. The concept of permutations is essential for understanding how to count and organize different sequences, especially when considering distinct groups of items. Permutations are closely related to the multiplication rule, as the number of ways to arrange objects can often be calculated by multiplying the number of choices available at each step.
Permutations: Permutations are arrangements of objects in a specific order. The order of the objects is crucial and changing the order creates a different permutation.
Probability: Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It connects various mathematical concepts by providing a framework to assess and quantify uncertainty in different scenarios, helping to determine outcomes based on different arrangements, selections, and occurrences.
Set theory: Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It provides a foundational framework for various mathematical concepts and operations, including relationships between different groups, classifications, and how elements interact within those groups. This framework is crucial for understanding concepts like subsets, Venn diagrams, and various set operations, which are fundamental in both theoretical and applied mathematics.
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