Glossary

3.3 Combinations and binomial coefficients

3 min read•july 30, 2024

Combinations and binomial coefficients are crucial tools in counting techniques. They help us count selections where order doesn't matter, like picking ice cream flavors or poker hands. These concepts are key to understanding probability and statistics.

Binomial coefficients, often written as "n choose k," show up in algebra and probability. They're the building blocks of and play a big role in expanding binomial expressions. Mastering these ideas opens doors to solving complex counting problems.

Combinations vs Permutations

Defining Combinations and Permutations

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  • Combinations select objects from a set without considering order
  • Permutations select objects from a set while considering order
  • Number of combinations always less than or equal to number of permutations for same set
  • Combinations used when selecting subset of items without replacement and without regard to order
  • Notation for combinations C(n,r) or (n choose r) represents n total objects and r chosen objects
  • Selecting ABC equivalent to selecting CBA in combinations, distinct in permutations

Applications and Examples

  • Combinations apply in probability theory, statistics, and combinatorics
  • Example: Selecting 3 flavors for an ice cream sundae from 8 options (order doesn't matter)
  • Example: Choosing 5 cards from a standard 52-card deck to form a poker hand
  • Permutations example: Arranging 5 books on a shelf (order matters)
  • Combinations example: Selecting 5 books to pack for a trip (order doesn't matter)

Combinations Formula

Formula and Derivation

  • formula C(n,r) = n! / (r! * (n-r)!)
  • n represents total objects, r represents chosen objects
  • Derived from permutation formula by dividing by r! to eliminate order effect
  • Symmetry property C(n,r) = C(n,n-r) demonstrates formula flexibility
  • Simplified cases C(n,1) = n and C(n,n) = 1 showcase formula variations

Calculation Techniques and Examples

  • Large factorials often require special techniques or calculators
  • Example: C(10,3) = 10! / (3! * 7!) = 120
  • Example: C(20,5) = 20! / (5! * 15!) = 15,504
  • Combination formula extends to binomial probability distributions
  • Example: Probability of exactly 3 heads in 5 coin flips = C(5,3) * (0.5)^3 * (0.5)^2

Combinations and Binomial Coefficients

Relationship and Notation

  • Binomial coefficients equivalent to combinations
  • Denoted as (n choose k) or C(n,k)
  • (n choose k) represents coefficient of x^k in (1+x)^n expansion
  • Calculated using combination formula (n choose k) = n! / (k! * (n-k)!)
  • Fundamental in probability theory, particularly
  • Example: Expanding (x + y)^4 using binomial coefficients

Applications in Mathematics

  • Binomial coefficients appear in algebra, probability, and combinatorics
  • Properties of binomial coefficients correspond to combination properties
  • Example: Number of ways to select 3 items from 7 = (7 choose 3) = 35
  • Example: Probability of exactly 2 successes in 5 trials with p=0.3 = (5 choose 2) * 0.3^2 * 0.7^3

Binomial Coefficient Properties

Pascal's Triangle and Its Properties

  • Pascal's triangle displays binomial coefficients in triangular array
  • Each number sums two numbers directly above, reflecting recursive property
  • Symmetry property visually represented in each symmetric row
  • Entries in nth row sum to 2^n, corresponding to total subsets of n-element set
  • Pascal's triangle determines binomial coefficients without direct calculation
  • Example: 5th row of Pascal's triangle 1, 5, 10, 10, 5, 1 represents (5 choose k) for k = 0 to 5

Problem-Solving Applications

  • Hockey stick identity and Fibonacci number relationship solve complex problems
  • Essential for solving probability distribution problems (binomial and negative binomial)
  • Example: Using Pascal's triangle to find C(10,4) without calculation
  • Example: Applying hockey stick identity to solve C(n,k) + C(n,k+1) = C(n+1,k+1)

Key Terms to Review (15)

Binomial Coefficient: The binomial coefficient, often denoted as $$C(n, k)$$ or $$\binom{n}{k}$$, 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 deeply connected to counting combinations, where it plays a critical role in determining how many ways a certain number of successes can occur in a given number of trials. It also forms the foundation for understanding the binomial distribution, which deals with scenarios involving two possible outcomes across multiple independent trials.
Binomial Distribution: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is crucial for analyzing situations where there are two outcomes, like success or failure, and is directly connected to various concepts such as discrete random variables and probability mass functions.
C(n, k): c(n, k) represents the number of combinations of n items taken k at a time, often referred to as 'n choose k'. This notation is crucial in understanding how to count selections where the order of selection does not matter, connecting deeply with the ideas of counting principles and binomial coefficients. Combinations are fundamental when determining probabilities in various scenarios, as they allow us to find how many ways we can choose subsets from a larger set without considering the arrangement of those subsets.
Choosing Teams: Choosing teams refers to the process of selecting a specific group of individuals from a larger pool, where the order of selection does not matter. This concept is crucial in probability as it connects to combinations and binomial coefficients, enabling us to calculate the number of ways to form teams under various conditions. By understanding how to count these selections, we can solve real-world problems that involve grouping people for competitions, projects, or activities without regard for the sequence in which they were chosen.
Combination: A combination is a selection of items from a larger set where the order of selection does not matter. This concept is crucial in probability and statistics, particularly when determining how many ways a certain number of items can be chosen from a given set, leading to the understanding of binomial coefficients, which express combinations mathematically.
Counting Principle: The counting principle is a fundamental concept in combinatorics that provides a systematic way to count the number of ways certain events can occur. It states that if one event can occur in 'm' ways and a second independent event can occur in 'n' ways, then the two events can occur together in 'm × n' ways. This principle lays the groundwork for understanding combinations, binomial coefficients, and how they relate to probability.
Events: In probability, an event is a specific outcome or a set of outcomes from a random experiment. Events can be simple, involving a single outcome, or compound, consisting of multiple outcomes. Understanding events is crucial when working with combinations and binomial coefficients, as it helps in determining the likelihood of various scenarios occurring.
Lottery odds: Lottery odds refer to the probability of winning a lottery game, typically expressed as a ratio of the number of ways to win to the total number of possible outcomes. Understanding these odds is crucial for players as it highlights the chances they have of winning against the vast pool of potential combinations, which connects to fundamental counting principles, combinations, and how counting techniques apply in calculating probabilities.
N! / (k! (n-k)!): The expression $$\frac{n!}{k!(n-k)!}$$ represents the number of ways to choose a subset of $k$ elements from a larger set of $n$ elements, known as combinations. This formula is central to counting problems where the order of selection does not matter, highlighting its importance in probability and statistics, especially when dealing with binomial distributions and other related concepts.
NCr: nCr, also known as combinations, represents the number of ways to choose 'r' elements from a set of 'n' elements without regard to the order of selection. This concept is fundamental in combinatorics, and it’s closely related to binomial coefficients, which are used in the binomial theorem to expand expressions involving powers of binomials. Understanding nCr helps in calculating probabilities and making decisions in various scenarios where order does not matter.
Order Does Not Matter: Order does not matter refers to the concept where the arrangement or sequence of items is irrelevant when counting combinations. In various mathematical scenarios, particularly in combinatorics, this principle signifies that selecting items in different orders results in the same outcome, focusing on the group as a whole rather than individual arrangements. This idea is foundational when working with combinations, where the goal is to find subsets of a larger set without regard to order.
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, creating a pattern that illustrates various properties of combinations and binomial coefficients. This triangle not only provides a visual representation for calculating combinations but also has applications in probability theory, particularly in problems involving counting and distribution.
Permutation vs Combination: Permutation refers to the arrangement of objects in a specific order, while combination refers to the selection of objects without regard to the order. These concepts are foundational in understanding how different groupings and arrangements can be formed from a set of items, which plays a crucial role in various applications such as counting principles, probability calculations, and combinatorial mathematics.
Sample Space: A sample space is the set of all possible outcomes of a random experiment or event. Understanding the sample space is crucial as it provides a framework for determining probabilities and analyzing events, allowing us to categorize and assess various situations effectively.
Selection without replacement: Selection without replacement refers to the process of choosing items from a set where once an item is selected, it cannot be selected again. This concept is crucial in probability because it affects the total number of possible outcomes and the calculations associated with combinations and binomial coefficients. Understanding this process is essential when determining how many ways you can select items and calculating probabilities in various scenarios.
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