Glossary

2.3 Combinations without repetition

3 min read•july 30, 2024

Combinations without repetition are a crucial concept in counting problems. They help us determine how many ways we can select items from a group when order doesn't matter. This topic builds on permutations, but focuses on selection rather than arrangement.

Understanding combinations is key for solving real-world problems in fields like probability, statistics, and genetics. We'll explore how to calculate combinations using formulas and tools like , and compare them to permutations to grasp their unique properties.

Combinations and their characteristics

Definition and Key Properties

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  • Combinations select objects from a set without regard to order
  • Each object can be selected only once, with
  • Number of combinations always less than or equal to permutations for the same set
  • Used to determine ways to select a from a larger set without order consideration
  • Notation represented as C(n,r) or ()
    • n represents total number of objects
    • r represents number of objects chosen
  • Form the basis for many advanced concepts in probability theory, statistics, and combinatorics

Examples and Applications

  • Selecting a committee of 3 people from a group of 10 (120 possible combinations)
  • Choosing 5 cards from a standard 52-card deck (2,598,960 possible combinations)
  • Picking winning lottery numbers (6 numbers from 49 in many lotteries)
  • Fundamental in calculating probabilities of events in games of chance
  • Used in biology for studying genetic combinations and population diversity
  • Applied in chemistry for molecular structure analysis and reaction possibilities

Combinations using binomial coefficients

Calculating Combinations

  • C(n,r) or (n choose r) represents number of ways to choose r objects from n objects
  • Formula for calculating combinations C(n,r)=n!r!(n−r)!C(n,r) = \frac{n!}{r!(n-r)!}
  • Binomial coefficient symmetric property C(n,r)=C(n,n−r)C(n,r) = C(n, n-r)
  • Can be expressed as a product of factors C(n,r)=n∗(n−1)∗...∗(n−r+1)r!C(n,r) = \frac{n * (n-1) * ... * (n-r+1)}{r!}
  • For large n and r values, use logarithms or specialized algorithms to avoid overflow errors

Pascal's Triangle and Binomial Coefficients

  • Pascal's Triangle generates binomial coefficients
  • Each number sum of two numbers directly above it
  • Rows of Pascal's Triangle correspond to n in C(n,r)
  • Positions in each row correspond to r
  • Example: Row 4 of Pascal's Triangle (1, 4, 6, 4, 1)
    • Represents C(4,0), C(4,1), C(4,2), C(4,3), C(4,4)
  • Useful for quickly finding small binomial coefficients

Permutations vs Combinations

Key Differences

  • Permutations involve arrangements where order matters
  • Combinations involve selections where order doesn't matter
  • Number of permutations always greater than or equal to combinations for same set
  • Permutation formula includes in numerator only
  • formula has factorials in both numerator and denominator
  • Relationship between permutations and combinations P(n,r)=C(n,r)∗r!P(n,r) = C(n,r) * r!

Problem-Solving Indicators

  • Key phrases in word problems indicate permutation or combination
  • Permutation indicators
    • "arrange" (arrange books on a shelf)
    • "order" (order of finish in a race)
    • "sequence" (possible PIN combinations)
  • Combination indicators
    • "select" (select committee members)
    • "choose" (choose lottery numbers)
    • "group" (group students for a project)
  • Example scenarios
    • Permutation: Arranging 5 people in a line (120 ways)
    • Combination: Selecting 3 people from a group of 5 for a committee (10 ways)

Applications of combinations

Probability and Statistics

  • Calculate number of ways to select a sample from a population
  • Fundamental in calculating binomial probabilities
    • Number of ways to choose successes from fixed number of trials
  • Used in survey sampling, experimental design, and hypothesis testing
  • Essential for polynomial expansion coefficients using binomial theorem
  • Applied in calculating probabilities in games of chance (poker hands, lottery odds)
  • Used in statistical inference and confidence interval calculations

Other Fields and Applications

  • Computer Science
    • Algorithms for generating all possible subsets of a set
    • Applications in cryptography and optimization problems
    • Used in analysis of algorithm complexity
  • Genetics
    • Calculate probability of inheriting certain traits
    • Study allele frequencies in population genetics
    • Analyze genetic diversity and mutation possibilities
  • Chemistry
    • Determine possible molecular structures
    • Analyze potential reaction pathways
  • Economics
    • Portfolio selection in financial mathematics
    • Game theory and strategic decision-making models

Key Terms to Review (16)

Arrangement vs Selection: Arrangement and selection are two fundamental concepts in combinatorics that deal with grouping objects. Arrangement refers to the different ways in which a set of items can be ordered, while selection focuses on choosing a subset of items from a larger set without regard to the order. Understanding the difference is crucial when solving problems related to combinations without repetition, as it helps clarify whether the focus is on how many ways items can be arranged or simply chosen.
Binomial Coefficient: The binomial coefficient, often denoted as $$\binom{n}{k}$$, represents the number of ways to choose a subset of size $$k$$ from a larger set of size $$n$$ without regard to the order of selection. This concept is foundational in combinatorics, linking counting principles to polynomial expansions and providing tools for solving various combinatorial problems. Understanding binomial coefficients is essential for comprehending how they appear in the Binomial Theorem, applications in counting problems, and their role in statistical inference.
C(n, r): c(n, r) represents the number of combinations of n items taken r at a time without repetition. This notation is used to calculate how many ways you can choose a subset of r items from a larger set of n items where the order does not matter, and each item can only be selected once. Understanding this concept is essential for solving problems related to selections and group formations in combinatorial mathematics.
Combination: A combination refers to a selection of items from a larger set, where the order of selection does not matter. This concept is foundational in various mathematical principles, particularly in counting techniques and arrangements, as well as in the analysis of structured designs. Understanding combinations allows for the exploration of patterns and arrangements, making it easier to analyze complex configurations and derive meaningful conclusions.
Factorial: A factorial, denoted as $$n!$$, is the product of all positive integers from 1 to n. It represents the number of ways to arrange n distinct objects and is foundational in counting principles, permutations, combinations, and other areas of combinatorics.
Lottery Problems: Lottery problems involve selecting a certain number of items from a larger set where each item can only be chosen once. These problems often require the use of combinations without repetition because the order in which the items are chosen does not matter, and each item can be selected only once. This concept is crucial in calculating probabilities and understanding the odds in various scenarios such as games of chance.
Multinomial Theorem: The multinomial theorem is a generalization of the binomial theorem that describes how to expand expressions of the form $$(x_1 + x_2 + ... + x_k)^n$$ into a sum involving products of the variables raised to various powers. This theorem connects to combinations and probabilities, providing a systematic way to find coefficients when distributing terms across multiple variables, as well as understanding the distributions of outcomes in scenarios with multiple categories.
N choose r: The term 'n choose r' refers to the mathematical concept of combinations, which calculates the number of ways to select 'r' items from a set of 'n' distinct items without regard to the order of selection. This concept is crucial when we want to determine how many different groups or subsets can be formed from a larger collection, emphasizing the idea that the arrangement of selected items does not matter.
NCr: nCr, or 'n choose r', represents the number of ways to choose r elements from a set of n elements without considering the order of selection. This mathematical concept is crucial for calculating combinations, especially in situations where the arrangement of selected items doesn't matter, as is often the case in probability and combinatorial problems.
No Repetitions Allowed: No repetitions allowed refers to a fundamental rule in combinations without repetition, where each selected element is unique and cannot be chosen more than once. This concept emphasizes the need for distinctness in selections, making it crucial for understanding how to calculate the total number of possible combinations when forming groups from a larger set. It plays a significant role in problems where order does not matter, and only unique groupings are considered.
Order does not matter: The phrase 'order does not matter' refers to a fundamental principle in combinatorics where the arrangement of selected items is irrelevant when forming groups or subsets. This concept is crucial for distinguishing between combinations and permutations, where combinations involve selection without regard to the sequence of items, while permutations consider the order. Recognizing that the arrangement of elements is not significant allows for a clearer understanding of how to calculate and analyze groupings effectively.
Pascal's Triangle: Pascal's Triangle is a triangular array of numbers that represents the coefficients of the binomial expansion. Each number is the sum of the two directly above it, showcasing a fascinating relationship between combinatorics and algebra. This triangle connects deeply with various concepts, such as counting combinations, understanding properties of binomial coefficients, and providing a visual representation of polynomial expansions through the Binomial Theorem.
Permutation vs Combination: Permutation refers to the arrangement of items in a specific order, while combination focuses on the selection of items without regard to the order. In situations where order matters, we use permutations; when only the selection is important, we use combinations. Understanding the difference is crucial, especially when analyzing scenarios with or without repetition.
Principle of Inclusion-Exclusion: The principle of inclusion-exclusion is a counting technique used to determine the number of elements in the union of several sets by considering the sizes of the individual sets and their intersections. This principle helps avoid overcounting by systematically adding and subtracting the sizes of various combinations of these sets. It is particularly useful in solving complex counting problems, generalizing to various scenarios, and understanding combinations without repetition.
Subset: A subset is a collection of elements that are all contained within another set. The concept of subsets is fundamental in combinatorics, as it allows for the exploration of how different combinations of elements can be formed without repetitions. Understanding subsets is crucial for analyzing how selections can be made from a larger group, leading to the study of combinations without repetition, where the order of selection does not matter.
Team Selection: Team selection refers to the process of choosing members from a larger group to form a smaller, cohesive team. This process is crucial when specific skills or attributes are required, and it often emphasizes combinations of individuals without repeating selections, ensuring that each member contributes uniquely to the team's objectives.
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