Problem 5 - College Algebra

Question

For the following exercises, evaluate the binomial coefficient. $$ \left(\begin{array}{l} 6 \\ 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The binomial coefficient is 15.

Step 1: Understand the Binomial Coefficient

The binomial coefficient \( \left( \begin{array}{c} n \ r \end{array} \right) \) represents the number of ways to choose \( r \) elements from a set of \( n \) elements without regard to order. It is calculated using the formula: \( \frac{n!}{r!(n-r)!} \).

Step 2: Substitute Values into the Formula

For \( n = 6 \) and \( r = 2 \), substitute these values into the formula: \( \left( \begin{array}{c} 6 \ 2 \end{array} \right) = \frac{6!}{2!(6-2)!} \).

Step 3: Calculate Factorials

First, calculate the factorials: - \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)- \( 2! = 2 \times 1 = 2 \)- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Step 4: Substitute Factorials into the Formula

Substitute the calculated factorials back into the formula:\( \frac{6!}{2!4!} = \frac{720}{2 \times 24} \).

Step 5: Simplify the Expression

Calculate the value: - First multiply \( 2 \times 24 = 48 \)- Divide the factorial result \( \frac{720}{48} = 15 \)

Step 6: Conclude the Result

The binomial coefficient \( \left( \begin{array}{c} 6 \ 2 \end{array} \right) \) evaluates to 15.

Key Concepts

Factorials

Factorials play a crucial role when dealing with binomial coefficients. In mathematics, a factorial is a product of all positive integers less than or equal to a given positive integer. It is denoted by an exclamation point (!). For example, the factorial of 6, written as 6!, is calculated as follows:

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

In this context, the factorial helps us simplify the expressions used in binomial coefficients and other combinatorial calculations. A smaller example is 2!, which equals 2 × 1 = 2.
Understanding and calculating factorials accurately can significantly ease the computation of more complex expressions encountered in combinatorics.

Combinatorics

Combinatorics is the branch of mathematics that focuses on counting, arrangement, and combination of objects. The binomial coefficient is a fundamental concept in combinatorics. It allows us to determine how many ways we can choose a subset of elements from a set, without considering the sequence or order of the elements.
The formula for the binomial coefficient when choosing \( r \) elements from \( n \) elements is given as:

• \( \left( \begin{array}{c} n \ r \end{array} \right) = \frac{n!}{r!(n-r)!} \)

This formula provides an efficient way to calculate large numbers of combinations without listing all possibilities, which is particularly useful when dealing with large sets.
Combinatorics appears in many aspects of applied mathematics, including probability theory and algorithm design.

Mathematical Formulas

Mathematical formulas are essential tools for expressing relationships and calculations in a compact form. In the exercise about binomial coefficients, the formula \( \frac{n!}{r!(n-r)!} \) is used to simplify understanding and computation of how elements can be combined.
This formula relies on factorial expressions, which account for permutations of elements, while the denominator adjusts for the cases where order does not matter.
Substituting values into a formula allows for straightforward calculation of even complex expressions by following structured steps. Here's how values are substituted in:

• For \( n = 6 \) and \( r = 2 \), we find \( \left( \begin{array}{c} 6 \ 2 \end{array} \right) = \frac{6!}{2!4!} \)

By managing the order of operations and leveraging properties of numbers, formulas like this one simplify potentially taxing arithmetic into manageable steps.