Problem 5 - Algebra and Trigonometry

Question

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l} {6} \\ {6} \end{array}\right) $$

Step-by-Step Solution

Answer

The value of the binomial coefficient \(\binom{6}{6}\) is 1.

Step 1: Identify the values of \(n\) and \(k\)

In this problem, the binomial coefficient is \(\binom{6}{6}\). This means that \(n = 6\) and \(k = 6\). The problem asks to find out how many ways there are to choose 6 items from a set of 6.

Step 2: Apply the property of binomial coefficient

By one of the properties of the binomial coefficients, it's known that \(\binom{n}{n} = 1 \). Hence, \(\binom{6}{6} = 1\) which means there is only one way to select all the 6 items from a set of 6 items.

Key Concepts

Combinatorics

Combinatorics is an area of mathematics focused on counting, arranging, and grouping objects. It deals with questions about how to count and organize different configurations of a set. In the context of our problem, combinatorics helps us understand the concept of binomial coefficients, which involve selecting a certain number of items from a larger set.
If you've ever wondered how many different ways you can arrange books on a shelf or select team members from a group, you're thinking about combinatorics!
The binomial coefficient is a practical tool in combinatorics for solving these problems, represented as \( \binom{n}{k} \), where \( n \) is the total number of items and \( k \) is the number of items to choose. This gives you a direct way to calculate the number of combinations possible.

Factorial Notation

Factorial notation is another key concept in combinatorics that simplifies counting. The factorial of a number \( n \), written as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In our binomial coefficient formula, factorial notation is used to find out how many ways you can arrange \( n \) items:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

For our specific problem \( \binom{6}{6} \), it simplifies mathematically as:

• \( \frac{6!}{6!(6-6)!} = \frac{6!}{6! \times 0!} \)

Since \( 0! = 1 \), this evaluates to 1, confirming there is just one way to select all items from the set.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Unlike continuous mathematics, which involves calculus and infinite numbers, discrete mathematics focuses on topics like integers, graphs, and logical statements.
Combinatorics, and thus binomial coefficients, are a vital part of discrete mathematics because they involve counting combinations and arrangements in finite sets. Understanding concepts like those explored in our exercise can help solve complex problems in computer science, cryptography, and network modeling.
The problem \( \binom{6}{6} \) is a classic example of how discrete mathematics can be applied to real-world scenarios where only whole numbers and specific selections are considered.