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Chapter 6: Problem 8

What is the coefficient of \(x^{8} y^{9}\) in the expansion of \((3 x+2 y)^{17} ?\)

Short Answer

Expert verified
The coefficient is 81,533,491,200.

Step by step solution

01

Identify the problem

Determine the coefficient of the term containing both \( x^8 \) and \( y^9 \) in the binomial expansion of \((3x + 2y)^{17}\).
02

- Recognize the Binomial Theorem

The Binomial Theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In this problem, \((a + b) = (3x + 2y)\) and \(n = 17\).
03

- Match the exponents

Identify the exponents for \(x\) and \(y\). The term we are looking for is \(x^{8} y^{9}\). According to the binomial expansion, this term corresponds to \[a^{n-k} b^k\] where \(3x)^{n-k}\) and \((2y)^k\), with \(a=3x\) and \(b=2y\).
04

- Relate exponents to indices

For the exponents to match, set the equations: \(n - k = 8\) and \(k = 9\). Solving yields: \(n = 17, n - k = 8\). So, \(k = 9\)
05

- Compute the coefficient

The coefficient of the term \((3x)^{n-k} (2y)^k\) in the expansion is \[\binom{17}{9} (3)^{8} (2)^{9}\]. Calculating each part: 1. Compute the binomial coefficient \( \binom{17}{9} \)2. Compute \( 3^8 \)3. Compute \( 2^9 \)
06

- Complete the calculations

1. Binomial coefficient: \(\binom{17}{9} = 24310\)2. \( 3^8 = 6561 \)3. \( 2^9 = 512 \)Combining all parts: \( \text{Coefficient} = 24310 \times 6561 \times 512 \). Calculating the final value, \( \text{Coefficient} = 81,533,491,200 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficient
The binomial coefficient is a crucial concept in the binomial theorem. In general, the binomial coefficient for given values of n and k is represented as \(\binom{n}{k}\). This symbol reads 'n choose k' and calculates how many ways you can choose k elements from a set of n elements.
The formula to calculate the binomial coefficient is: \[ \binom{n}{k} = \frac{n!}{k! (n - k)!} \] where \(!\) denotes the factorial function.
For the exercise at hand, we need \(\binom{17}{9}\), which can be calculated as: \[ \binom{17}{9} = \frac{17!}{9! (17 - 9)!} = \frac{17!}{9! \times 8!} \] Using a calculator can simplify this step, leading to the value \(\binom{17}{9} = 24310\).
Polynomial Expansion
Polynomial expansion using the binomial theorem helps us expand expressions of the form \((a + b)^n\). The binomial theorem states: \[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \] Each term in this expansion is a product of a binomial coefficient, a power of \(a\), and a power of \(b\).
In our problem, we need to deal with the expansion of \((3x + 2y)^{17}\). This means: \[ (3x + 2y)^{17} = \binom{17}{k} (3x)^{17-k} (2y)^k \] We are particularly interested in terms where we end up with \(x^8 y^9\).
Exponent Matching
Exponent matching is a technique used to find the specific term in the polynomial expansion that matches the required exponents. In this exercise, we want the term with \(x^8 y^9\).
From the binomial expansion formula, we match exponents: \[ (3x)^{17-k} (2y)^k \] To find the corresponding term, solve the exponent equations: \[ 17 - k = 8 \] \[ k = 9 \] Once you solve these, you find that \(k = 9\). Hence, \(17 - k = 8\), perfectly matching the exponents of \(x\) and \(y\).
Coefficient Calculation
Once the exponent matching is done, the next step is to calculate the coefficient of the term. The general term includes the binomial coefficient and powers of the constants. The term is: \[ \binom{17}{9} (3)^{8} (2)^{9} \] Breaking it down step by step:
  • Binomial Coefficient: \(\binom{17}{9} = 24310\)
  • Power of 3: \(3^8 = 6561\)
  • Power of 2: \(2^9 = 512\)
Now, combine all these factors: \[ \text{Coefficient} = 24310 \times 6561 \times 512 \] This leads to the final coefficient of \(81,533,491,200\).
This is the coefficient of the term \(x^8 y^9\) in the expansion of \((3x + 2y)^{17}\).

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