Problem 21 - Algebra 2

Question

Evaluate each expression. \(_{6} \mathrm{C}_{2}\)

Step-by-Step Solution

Answer

has a value of 15.

Step 1: Understanding Combination Notation

The notation has the general form , which represents the number of combinations of 'n' items taken 'r' at a time without regard to the order. It is calculated using the formula .

Step 2: Apply the Combination Formula

To evaluate , we apply the combination formula , which simplifies to .

Step 3: Solving the Factorials

Calculate the factorials in the expression to find . After that, divide the numerator by the denominator to get the answer.

Key Concepts

Combination Notation

Understanding combination notation is vital for mastering the basics of combinatorics, a branch of algebra that deals with counting combinations and permutations. Combination notation is expressed as \( _nC_r \), where \(n\) and \(r\) are non-negative integers and \(r \leq n\).

It signifies the number of ways to choose \(r\) elements from a larger set of \(n\) elements without considering the order of selection. It's an essential concept for problems involving probability, statistics, and various fields requiring combinatorial analysis.

Real-Life Application of Combination Notation

Imagine you have a box of six different colors of socks, and you need to pick two for a trip. You might wonder how many different pairs you can make. This is where combination notation comes into play, allowing you to calculate the total possibilities easily. Using the notation \( _6C_2 \), you could quickly find that there are 15 different pairs of socks possible.

To improve your grasp of combination notation, visualize the process as simply choosing 'r' items from a bigger 'n' item pool without worrying about the order they're picked in.

Factorials

Factorials are a fundamental concept in algebra, especially in the field of combinatorics. A factorial, denoted by the exclamation point (!), represents the product of all positive integers up to a given number. Specifically, the factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\), written as \(n!\).

For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). It's important to note that by definition, \(0! = 1\).

Why Factorials Matter

In the context of combinations, factorials are used to eliminate the repeated counts of arrangements. They let us handle large quantities systematically when arranging items or considering possible outcomes in probability problems. Mastering factorials simplifies dealing with large numbers and enables us to solve expressions involving 'n choose r' easily.

One way to better understand factorials is to practice writing out the first few values (1!, 2!, 3!, etc.) in expanded form to see the pattern inherent in the factorial operation. Moreover, practicing with calculators or software that can handle large factorial computations can aid deeper understanding and appreciation for how these numbers grow rapidly.

Binomial Coefficient

In algebra, the binomial coefficient is a cornerstone of combinatorial mathematics. It arises in the binomial theorem, which provides a way to expand expressions raised to a power. The binomial coefficient, written as \( _nC_r \) or \( \binom{n}{r} \), represents the coefficient of the \(x^{n-r}y^r\) term in the expansion of the binomial \( (x + y)^n \).

Mathematically, the binomial coefficient is defined using factorials: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). It calculates how many ways one can choose \(r\) elements out of a set of \(n\) without regard to order. Thus, it's another representation of combinations.

Understanding Through Patterns

One strategy to build a deeper understanding of binomial coefficients is to explore Pascal's Triangle, which neatly arranges these coefficients and shows their relationships. Each number in Pascal's Triangle is the sum of the two numbers directly above it, and these numbers directly correspond to binomial coefficients.

Binomial coefficients play a crucial role in many areas of mathematics, including algebra, probability, and statistics. They help us in quantifying the probabilities of various outcomes and are especially useful when working with probabilities of independent and mutually exclusive events.