Problem 15 - College Algebra

Question

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{5} C_{0} $$

Step-by-Step Solution

Answer

The value of \(_{5}C_{0}\) is 1.

Step 1: Identify 'n' and 'r'

In the binomial coefficient expression \(_{5}C_{0}\), 'n' is 5 and 'r' is 0.

Step 2: Apply the Binomial Coefficient Formula

The formula for the binomial coefficient is \(_{n}C_{r}= \frac{n!}{r!(n-r)!}\). To apply this formula, start by calculating the factorial of 'n', 'r' and 'n-r'. n! = 5! = 5 * 4 * 3 * 2 * 1. The factorial of 'r' which is 0! = 1 (by definition). The factorial of 'n-r' which would be (5-0)! = 5! = 5 * 4 * 3 * 2 * 1.

Step 3: Substitute Values into Formula

Substitute all calculated values into the formula: \(_{5}C_{0}= \frac{5!}{0!(5-0)!} = \frac{5!}{0!*5!}\). Due to the fact that 0! is 1, eliminate this term to simplify calculation.

Step 4: Simplify

After substituting, \(_{5}C_{0}= \frac{5!}{5!} = 1\). The two 5! terms cancel out.

Key Concepts

Understanding Factorials

A factorial, denoted as an exclamation mark (!), is a concept used in mathematics to indicate the product of all positive integers less than or equal to a given number. For example, the factorial of 5, written as 5!, equals 5 × 4 × 3 × 2 × 1, which totals 120.
Factorials are particularly important in the field of combinatorics and probability, as they help organize how different combinations or arrangements can be made.
By definition, the factorial of 0, which is 0!, equals 1. This might seem counterintuitive, but it's a fundamental rule that helps maintain consistency across various mathematical formulas, such as the binomial coefficient.
Here's a quick recap of how factorials function:

• 1! = 1
• 2! = 2 × 1
• 3! = 3 × 2 × 1
• n! = n × (n-1) × ... × 2 × 1

Understanding factorials is crucial for delving into more advanced mathematical concepts.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and analyzing different configurations of sets. It studies how to select or arrange objects under varying conditions and constraints.
At its core, combinatorics involves problems of combinations and permutations. Combinations deal with selecting items without considering the order, while permutations do consider the order.
When you encounter problems asking you to determine the number of ways to choose certain elements from a larger set, it’s typically about combinations. For example, the problem of selecting 2 colors from a palette of 5 colors involves combinations, since the order of selection does not matter.
The formula used for combinations is known as the binomial coefficient formula:

• \[_{n}C_{r} = \frac{n!}{r!(n-r)!}\]

In this formula:

• \( n \) represents the total number of items.
• \( r \) represents the number of items to choose.

Combinatorics helps us better understand data organization, structure, and make informed decisions based on available information.

Exploring the Binomial Theorem

The binomial theorem is a pivotal algebraic formula that specifies how to expand expressions raised to a power. It shows how to expand \((a + b)^n\) into a sum involving terms of the form \(a^{b}\) and \(b^{n-b}\), with the binomial coefficients serving as multipliers for each term.
This theorem is scientifically valuable as it helps break down complex polynomials into manageable parts, making them easier to solve or manipulate.
The general formulation of the binomial theorem is as follows:

• \[ (a + b)^n = \sum_{k=0}^{n} \, _{n}C_{k} \, a^{n-k} \, b^{k} \]

Each term in this expansion includes a binomial coefficient, \(_{n}C_{k}\), which determines how many ways \(k\) items can be chosen from \(n\) total items, reflecting the symmetry and balance of the theorem.
Understanding the binomial theorem properly enriches one's broader insight into polynomial algebra and shows the interconnectedness of mathematics through the use of simple yet powerful concepts like binomial coefficients and factorials.