Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}5 \\ 3\end{array}\right)\)

Step-by-Step Solution

Answer

The binomial coefficient \( \left( \begin{array}{l}5 \\ 3\end{array}\right) \) is 10.

Step 1: Understand Binomial Coefficient

The binomial coefficient \( \left( \begin{array}{c} n \ k \end{array} \right) \) represents the number of ways to choose \( k \) items from \( n \) items without regard to the order. The binomial coefficient formula is given by \( \left( \begin{array}{c} n \ k \end{array} \right) = \frac{n!}{k!(n-k)!} \).

Step 2: Plug Values into Formula

We are required to find \( \left( \begin{array}{c} 5 \ 3 \end{array} \right) \). Using the formula, we have \( \left( \begin{array}{c} 5 \ 3 \end{array} \right) = \frac{5!}{3!(5-3)!} \).

Step 3: Calculate Factorials

Calculate the factorials involved: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \), \( 3! = 3 \times 2 \times 1 = 6 \), and \( (5-3)! = 2! = 2 \times 1 = 2 \).

Step 4: Simplify the Expression

Substitute the factorial values into the expression from Step 2: \( \left( \begin{array}{c} 5 \ 3 \end{array} \right) = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \).

Key Concepts

Factorials

A factorial, denoted as \( n! \), is a mathematical operation that plays a crucial role in permutations and combinations. It involves multiplying a series of descending natural numbers. For example, \( 5! \) means \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials are foundational in calculating binomial coefficients, where they help determine the number of ways to arrange or choose items from a set. In our context, factorials calculate the possible ways to choose items by dividing them into groups. This mechanism simplifies complex combinatorial problems into manageable multiplication and division tasks. When solving binomial coefficient problems, understanding factorials helps break down the combinations into straightforward arithmetic.

Combinatorics

Combinatorics is the branch of mathematics focused on counting and arrangements. It's all about exploring how to select, arrange, and combine objects within a set.
When it comes to binomial coefficients, combinatorics teaches us how to determine the number of ways to choose \( k \) items from \( n \) items. In our exercise, we're interested in \( \binom{5}{3} \), which is the number of combinations for selecting 3 items from 5 without considering order.
This is expressed using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This formula accounts for all possible arrangements and then manages redundancies to give precise counts. Thus, understanding basic combinatorics, allows us to answer questions about possible configurations efficiently.

Mathematical Notation

Mathematical notation is a system of symbols used to express mathematical ideas concisely. In the context of binomial coefficients, the notation \( \binom{n}{k} \) represents the number of ways to choose \( k \) items from \( n \) items.
This notation is powerful and succinct, conveying complex concepts through simple symbols. The expression abbreviates a process involving factorials and division, saving time and reducing confusion.
Proper understanding of this notation helps in decoding many problems in statistics and probability, ensuring clarity in problem-solving. By recognizing how these symbols work, you can seamlessly follow and solve combinatorial problems like choosing representatives from a committee or creating a k-member team.