Problem 59 - Precalculus with Limits

Question

In Exercises 53 - 60, find the coefficient of the term in the expansion of the binomial. Binomial \( \quad \quad \quad \) Term \( \left(x^2 + y\right)^{10} \quad \quad \quad ax^8y^6 \)

Step-by-Step Solution

Answer

The coefficient of the term \(x^8y^6\) in the expansion of the binomial \(\left( x^2 + y \right)^{10}\) is 210.

Step 1: Set up of Required Term

Identify terms in the binomial. For the term \(x^8y^6\), in \((x^2+y)^{10}\), \(x^2\) can be treated as 'a' and 'y' as 'b'. The powers of 'a' and 'b' are 4 and 6 respectively (since \(x^8\) can be written as \((x^2)^4\)). Hence, the term from the binomial theorem equation is represented as \(\binom{n}{r} \cdot (a)^{n-r} \cdot (b)^r\), where n=10, r=6, a=x^2, b=y.

Step 2: Calculation of the Binomial Coefficient

Calculate \(\binom{n}{r}\), which is \( \binom{10}{6} \). This can be calculated as \( \frac{10!}{6!(10-6)!} \), which equals to 210.

Step 3: Substitute the Values

Substitute the values of \(n\), \(r\), \(a\), \(b\) and \( \binom{n}{r} \) into the equation. This gives us \( \binom{10}{6} \cdot (x^2)^{10-6} \cdot (y)^6 \) = 210 \cdot x^8 \cdot y^6.

Key Concepts

Binomial Expansion

Binomial expansion is a crucial concept in algebra that allows us to expand expressions raised to a power. Using the binomial theorem, we can expand expressions like \((a + b)^n\) into a sum of terms involving coefficients, powers of \(a\), and powers of \(b\). The general formula is:

• \((a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\)

This formula means we are summing up terms where each term is formed by multiplying a binomial coefficient, a power of \(a\), and a power of \(b\). The exponents of \(a\) and \(b\) always add up to \(n\), which is the original power the binomial is raised to. This expansion is particularly useful when dealing with powers larger than 2, as manually multiplying can be cumbersome.
For instance, in the expression \((x^2 + y)^{10}\), we use the binomial theorem to find the term where \(x^8y^6\) appears, employing the structure of the expansion formula to guide us.

Coefficient Calculation

Calculating the coefficient of a specific term in a binomial expansion involves using binomial coefficients, which are denoted as \(\binom{n}{r}\). These coefficients determine how many ways we can pick \(r\) elements from \(n\) without considering order. The formula for calculating a binomial coefficient is:

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

Here, \("!"\) represents the factorial operation, which is the product of an integer and all the positive integers below it. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
In our example, we calculated \(\binom{10}{6}\), which equates to 210. This value of 210 is the coefficient of the term \(x^8y^6\) in the expansion of \((x^2+y)^{10}\). Coefficients are essential in determining the actual numeric value of each term in a polynomial expansion.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations, and it's vital in understanding binomial expansions. It helps us determine how terms are selected and arranged. Binomial coefficients (\(\binom{n}{r}\)), a key component of combinatorics, represent the number of combinations we can make.

• They are derived using the formula: \(\binom{n}{r} = \frac{n!}{r!(n-r)!} \)
• They tell us how many distinct ways we can choose \(r\) items from a set of \(n\) items.

In the context of our exercise, combinatorics helps figure out how many terms in the expanded form are needed and their coefficients, like in the term \(x^8y^6\). Understanding combinatorics enhances our ability to solve problems involving probability and statistics, as these fields often require counting arrangements or selections of datasets.