Problem 50 - Algebra and Trigonometry

Question

Simplify using the Binomial Theorem. Show that \(\left(\begin{array}{l}{n} \\ {0}\end{array}\right)=1\) and \(\left(\begin{array}{l}{n} \\ {n}\end{array}\right)=1\)

Step-by-Step Solution

Answer

Both \(\binom{n}{0}\) and \(\binom{n}{n}\) evaluate to 1.

Step 1: Introduce the Binomial Theorem

The Binomial Theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is the binomial coefficient and represents the number of ways to choose \(k\) elements from \(n\).

Step 2: Evaluate \(\binom{n}{0}\)

According to the binomial coefficient formula, \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Setting \(k = 0\), we get \(\binom{n}{0} = \frac{n!}{0!(n-0)!} = \frac{n!}{n!} = 1\). This confirms that there is one way to choose zero elements from \(n\) elements, which is to choose nothing.

Step 3: Evaluate \(\binom{n}{n}\)

Using the formula again, \(\binom{n}{n} = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \, 0!} = \frac{n!}{n!} = 1\). This means there is exactly one way to choose all \(n\) elements from \(n\) elements, which is to choose everything.

Step 4: Conclusion

Thus \(\binom{n}{0} = 1\) and \(\binom{n}{n} = 1\), confirming the problem statement using the Binomial Theorem.

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are essential in algebra, especially when you dive into the Binomial Theorem. They tell us how many different ways we can choose a subset of elements from a larger set. The notation \( \binom{n}{k} \) is used to represent these coefficients, where \( n \) is the total number of elements, and \( k \) is the number of elements we're choosing. This is often read as 'n choose k'.
A binomial coefficient can be calculated using the formula:

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

Here, \( ! \) denotes factorial, which we'll explain in the next section. The essence of the binomial coefficient is its ability to calculate the number of ways to select \( k \) elements from \( n \) elements, without regard to the order of selection.
For example, \( \binom{5}{2} \) would calculate the number of ways you can pick 2 objects from a set of 5. Interestingly, certain binomial coefficients like \( \binom{n}{0} \) and \( \binom{n}{n} \) always equal 1. This happens because there's exactly one way to choose none (or all) of the elements in a set.

Exploring Factorial Notation

Factorial notation is a simple yet powerful tool, especially in combinatorics and calculations involving binomial coefficients. The factorial of a number \( n \) is denoted \( n! \), and it represents the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

• This concept is fundamental because it simplifies the calculation of permutations and combinations.
• An important note is that \( 0! = 1 \) by definition, which is crucial when calculating binomial coefficients like \( \binom{n}{0} \)

The factorial function grows very quickly, which means it becomes crucial for calculations in algorithms and complex expressions. For binomial coefficients, the use of factorial makes computations straightforward and intuitive, without needing to manually count combinations.

Basics of Combinatorics in Algebra

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It's like the math of choices and has a wide range of applications from simple problems such as choosing pizza toppings to more complex areas like cryptography and network theory.
In the context of the Binomial Theorem, combinatorics forms the backbone of understanding how terms in the expansion are formed. The Binomial Theorem provides a way to expand expressions raised to a power, using combinatorial methods to handle the arrangement of terms. This is where binomial coefficients step in, showing how each term in the expansion is a unique arrangement of components.
With combinatorics, students can better understand the logic behind permutations and combinations, making sense of how larger sets can be broken down into smaller, manageable tasks and results. Additionally, this provides insights into problem-solving and analytical thinking, crucial skills in both academic and real-world scenarios.