Problem 21 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$C(6,0)$$

Step-by-Step Solution

Answer

\( C(6, 0) = 1 \).

Step 1: Understanding the Notation

The expression \( C(6,0) \) represents a combination. A combination is a way to select items from a group, where the order does not matter. Here, we're selecting \(0\) items from a total of \(6\) items.

Step 2: Combination Formula

The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose. In this case, \( n = 6 \) and \( r = 0 \).

Step 3: Substituting Values

Substitute the known values into the formula: \[ C(6, 0) = \frac{6!}{0! \times (6-0)!} \].

Step 4: Factorial Evaluation

Evaluate the factorials: \( 6! = 720 \), \( 0! = 1 \), and \( 6! = 720 \).

Step 5: Simplifying the Expression

Substitute the evaluated factorials back into the formula: \[ C(6, 0) = \frac{720}{1 \times 720} = 1 \].

Key Concepts

Factorials

Factorials are mathematical expressions that represent the product of all positive integers up to a certain number. They are denoted by an exclamation mark, like this: \( n! \). For example, \( 5! \) would be calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This concept is foundational in combinatorics, including topics such as permutations and combinations.

Some important points about factorials include:

• Factorials grow very quickly. Even \( 10! \) is already 3,628,800.
• By definition, \( 0! \) is equal to 1. This might seem counterintuitive at first, but it's useful for simplifying equations in discrete mathematics.

Factorials are instrumental in calculating combinations, which use them to determine different ways of selecting items from a group. Understanding the nature of factorials helps in simplifying complex expressions involving selections.

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics and is often represented as \( C(n, r) \) or \( \binom{n}{r} \). It helps calculate the number of ways to choose \( r \) items from \( n \) total items without considering the order. The formula to compute a binomial coefficient is \( \frac{n!}{r! (n-r)!} \).

The binomial coefficient has many applications:

• In statistics, it is used in binomial distributions to find probabilities.
• It appears in Pascal's Triangle, where each entry is a binomial coefficient relating to that row's indices.
• In algebra, it contributes to the binomial theorem, which expands expressions of the form \( (a + b)^n \).

The calculations of combinations using binomial coefficients are efficient because of the factorial division, which typically results in a simplification that cancels out complex terms.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with discrete (distinct and separate) objects. It includes topics such as graph theory, set theory, and combinatorics. Combinatorics, the study of counting, arrangement, and operation within a set, is a significant part of discrete mathematics.

Key topics in discrete mathematics relevant to combinations include:

• Counting principles, which provide methods to count the number of arrangements and selections.
• Graph theory, which uses discrete structures to model pairwise relations between objects.
• Set theory, focusing on the study of collections of objects, which underpins many combinatorial problems.

Discrete mathematics is immensely useful in computer science, cryptography, and logistics for optimizing processes and solving complex algorithmic problems. It provides a breadth of techniques for dealing with finite systems and structures, such as calculating combinations like \( C(6,0) \).