Problem 48 - College Algebra with Corequisite Support

Question

Find \(\left(\begin{array}{c}n \\\ k-1\end{array}\right)+\left(\begin{array}{l}n \\ k\end{array}\right)\) and write the answer as a binomial coefficient in the form \(\left(\begin{array}{l}n \\ k\end{array}\right)\). Prove it.

Step-by-Step Solution

Answer

\(\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}\) using Pascal's identity.

Step 1: Understand the Problem

We need to simplify the expression \( \binom{n}{k-1} + \binom{n}{k} \) using the properties of binomial coefficients. Our goal is to express this sum as a single binomial coefficient \( \binom{n+1}{k} \).

Step 2: Utilize Pascal's Identity

Recall Pascal's identity for binomial coefficients, which states that \( \binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k} \). This identity can help us rewrite the sum of two binomial coefficients in a simplified form.

Step 3: Apply Pascal's Identity

Apply Pascal's identity directly to the given expression: \( \binom{n}{k-1} + \binom{n}{k} \). By the identity, this is equal to \( \binom{n+1}{k} \).

Step 4: State the Answer

Using Pascal's identity, we have simplified the expression \( \binom{n}{k-1} + \binom{n}{k} \) to \( \binom{n+1}{k} \). Therefore, the given expression can be written as a single binomial coefficient: \( \binom{n+1}{k} \).

Key Concepts

Pascal's Identity

Pascal's Identity is a fundamental principle in combinatorics that helps simplify expressions involving binomial coefficients. It states that the sum of two specific binomial coefficients, \( \binom{n}{k-1} \) and \( \binom{n}{k} \), is equal to a single binomial coefficient, \( \binom{n+1}{k} \).
Pascal's Identity can be written as:

• \( \binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k} \).

This identity is derived from the arrangement of numbers in Pascal's Triangle. Each number in the triangle is the sum of the two numbers directly above it, which is a visual representation of Pascal's Identity.
To understand this concept, consider how combinations work:

• \( \binom{n}{k-1} \) represents choosing \( k-1 \) items from \( n \), and \( \binom{n}{k} \) represents choosing \( k \) items.

The identity elegantly expresses how these combinations relate, showing a relationship so cohesive in solving complex combinatorics.Shared in mathematical problems and exercises, Pascal's Identity not only serves to simplify expressions but also provides insightful proofs in algebra and combinatorics.

Combinatorics

Combinatorics is the branch of mathematics focusing on counting, selection, and arrangement of elements within a set. It answers questions like "How many ways can we choose certain items from a larger set?" or "What are the possible combinations?"
Central to combinatorics is the concept of binomial coefficients, represented as \( \binom{n}{k} \). This notation helps us calculate the number of ways to choose \( k \) items from \( n \) items without regard to order. The formula for a binomial coefficient is given by:

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

Where "\(!\)" denotes factorial, meaning the product of all positive integers up to that number.
In the broader scope of combinatorics, these coefficients are useful in understanding permutations, generating functions, and combinations in probability theory. The beauty of combinatorics lies in its ability to solve complex arrangements with simple foundational tools. Techniques like Pascal's Identity simplify these tasks by providing straightforward methods for calculating and verifying results.

Algebraic Proof

Algebraic Proof is a method of verifying mathematical statements using algebraic techniques and logical steps. In the context of binomial coefficients, algebraic proof involves demonstrating identities like Pascal's Identity through logical reasoning.
To prove Pascal's Identity, one can use an algebraic approach. Start by considering the combinatorial meaning:

• Think of a group of \( n+1 \) items where you want to choose \( k \).
• One way is to consider these items being chosen in two steps: first choosing one item which leaves \( n \) options and then choosing either \( k-1 \) or \( k \) items from the remaining \( n \) items.

Thus, the sum \( \binom{n}{k-1} + \binom{n}{k} \) becomes \( \binom{n+1}{k} \) as per the identity.
This proof is not only logical but also deeply satisfying as it leverages the fundamental properties of combinations. Algebraic proofs serve as a powerful tool in mathematics, giving profound insight into the relationships and properties of numbers and structures.