Problem 16 - Discrete Mathematics with Applications

Question

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{8}$$

Step-by-Step Solution

Answer

The largest binomial coefficient in the expansion of \((x+y)^8\) is \(\binom{8}{4} = \boxed{70}\).

Step 1: Identify the Binomial Coefficient

The binomial coefficient or combination is given by: $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ where \(n!\) is the factorial of \(n\) (i.e., the product of all positive integers less than or equal to \(n\)) and \(k\) is an integer between \(0\) and \(n\). In our problem, we have \(n = 8\), and we need to find the largest binomial coefficient for all values of \(k\) between \(0\) and \(8\).

Step 2: Find the Symmetry Property of the Binomial Coefficients

Binomial coefficients exhibit symmetry: \(\binom{n}{k} = \binom{n}{n-k}\). This means that we only need to evaluate the coefficients for \(k = 0\) to \(k = 4\), because the coefficients for \(k = 5\) to \(k = 8\) will have the same values as the first half.

Step 3: Calculate the Binomial Coefficients for k = 0 to 4

Calculate the binomial coefficients for the first half (i.e., for \(k = 0\) to \(k = 4\)): \[ \binom{8}{0} = \frac{8!}{0!(8-0)!} = 1 \\ \binom{8}{1} = \frac{8!}{1!(8-1)!} = 8 \\ \binom{8}{2} = \frac{8!}{2!(8-2)!} = 28 \\ \binom{8}{3} = \frac{8!}{3!(8-3)!} = 56 \\ \binom{8}{4} = \frac{8!}{4!(8-4)!} = 70 \]

Step 4: Find the Maximum Binomial Coefficient

From the calculated values, we can observe that the maximum binomial coefficient is \(\binom{8}{4} = \binom{8}{8-4} = 70\). Therefore, the largest binomial coefficient in the expansion of \((x+y)^8\) is \(\boxed{70}\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a foundational concept in algebra that describes how to expand expressions of the form \((x+y)^n\). This theorem states that any power of a binomial can be expressed as a sum of terms, each containing coefficients known as binomial coefficients.
These coefficients are represented as \( \binom{n}{k} \), which denotes the number of ways to choose \(k\) elements from a set of \(n\) elements without taking the order into account.
The expression \((x+y)^8 = \sum_{k=0}^{8} \binom{8}{k} x^{8-k} y^k\)exemplifies this expansion. Here, each term has a coefficient \( \binom{8}{k} \), representing different combinations of the components.

• Each coefficient tells you the number of ways to reach a particular term by choosing \(k\) elements.
• These numbers display symmetry and are integral to many areas of mathematics, including calculus and algebraic operations.

Factorial

A factorial, denoted by \(n!\), is a key operation when working with binomial coefficients and combinations. Factorials are the product of all positive integers less than or equal to a given number \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow very quickly and are foundational in calculating combinations and permutations.
In the binomial coefficient formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), factorials play the crucial role of determining the total number of possible selections (or arrangements) of a subset. Understanding factorials is vital for grasping how different elements can be arranged or selected in various mathematical scenarios.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting principles. It's used to determine the number of possible arrangements in a set. In our exercise, combinatorics helps decide how many ways there are to select \(k\) items from \(n\) items, which directly relates to binomial coefficients.
The idea of finding the largest coefficient in \((x+y)^8\) involves understanding how different selections are distributed within the expansion, highlighting the role of combinatorics in organizing and solving such problems.

• Combinatorics simplifies complex counting problems by using general principles to find solutions efficiently.
• It not only counts outcomes but also explores the structure of sets and their subsets.

Symmetry Property of Binomial Coefficients

The symmetry property of binomial coefficients is a fascinating aspect that states: \(\binom{n}{k} = \binom{n}{n-k}\). This property indicates that the binomial coefficients are mirror images around the center of the expansion set.This simplification allows half the computations in cases where symmetry can be applied.In our problem of expanding \((x+y)^8\), this means that once we calculate coefficients up to \(k=4\), we already know the values for \(k=5\) through \(k=8\).This symmetry doesn't just simplify calculations; it offers deep insights into the structural properties of polynomial expansions.

• This reflection property is not only intriguing but practical, reducing computation time significantly.
• Recognizing symmetry helps in solving a wide range of mathematical problems effectively.