Problem 6 - College Algebra

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 \( \binom{5}{3} \) is 10.

Step 1: Understand the Definition of the Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), is defined as the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of the elements. It is given by the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). In this problem, \( n = 5 \) and \( k = 3 \).

Step 2: Substitute the Values into the Formula

Substitute \( n = 5 \) and \( k = 3 \) into the formula for the binomial coefficient: \( \binom{5}{3} = \frac{5!}{3!(5-3)!} \).

Step 3: Simplify the Factorials

Calculate the factorial of each number: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \), \( 3! = 3 \times 2 \times 1 = 6 \), and \( 2! = 2 \times 1 = 2 \).

Step 4: Perform the Division

Substitute the factorial values back into the binomial coefficient expression: \( \frac{120}{6 \times 2} = \frac{120}{12} \).

Step 5: Calculate the Final Result

Divide 120 by 12 to get the final result: \( \frac{120}{12} = 10 \).

Key Concepts

Factorials

Factorials are incredibly important in mathematics, especially when dealing with permutations, combinations, and in this case, binomial coefficients. A factorial is denoted by an exclamation mark (!) and is the product of all positive integers up to a certain number. For instance, for any non-negative integer \( n \), \( n! \) is calculated as \( n \times (n-1) \times (n-2) \times \ldots \times 1 \). This can be particularly useful when evaluating expressions like binomial coefficients.

• \( 0! \) is defined to be 1. This is because the empty product (the product of no numbers at all) is always 1.
• Factorials grow very fast; even a seemingly small number like 10! equals 3,628,800.

Knowing how to compute factorials helps streamline solving problems involving combinations and arrangements.

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and combination of sets. It answers questions like, "How many ways can we arrange these items?"
In the context of the binomial coefficient, we are dealing with combinations, which tells us how many ways we can choose \( k \) items from \( n \) without consideration of the order.

• The binomial coefficient \( \binom{n}{k} \) is computed using factorials, allowing us to explore various combination scenarios in discrete mathematics and probability theory.
• It is crucial for understanding the concept of combinations where repetition is not allowed.

Combinatorics introduces you to a world of possibilities in arrangements and selections, forming the foundation for more advanced topics like graph theory and non-linear dynamics.

Mathematical Notation

Mathematical notation provides a universal language for expressing mathematical ideas precisely and succinctly. In combinatorics, a common notation is the binomial coefficient, represented as \( \binom{n}{k} \). This notation indicates the number of ways to choose \( k \) items from \( n \) items.
Let's break it down

• \( \binom{n}{k} \) is read as "n choose k," underlining its purpose in selecting items.
• This compact notation allows for easy communication of complex ideas.
• It's important to use correct notation for clarity and consistency in mathematical communication.

Mastering mathematical notation paves the way for better understanding in varied areas of math and helps in sharing solutions effectively with others.