Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Question 5: Binomial Expansion
**Question:**
What is the value of the coefficient of the 4th term in the expansion of \((a + b)^{12}\)?
**Options:**
- 985
- 1,320
- 220
- 445
**Explanation:**
To solve for the coefficient of the 4th term in the binomial expansion of \((a+b)^{12}\), we use the binomial theorem formula:
\[
(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For the 4th term, we need the term where \(k=3\) (note that binomial theorem terms are indexed starting from \(k=0\)). The binomial coefficient \(\binom{12}{3}\) determines the coefficient we're looking for. We can compute it as follows:
\[
\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3! \cdot 9!}
\]
This simplifies to:
\[
\frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]
Therefore, the correct answer is:
- **220**
Feel free to use this derived explanation along with the question and provided options for educational purposes, helping students understand the binomial expansion and coefficient calculation methods.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe20fc6e6-2ea7-4c1e-8366-712724b6961a%2Fbeb1e05a-c189-42e9-b0aa-dd610f523f93%2F140zmvj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 5: Binomial Expansion
**Question:**
What is the value of the coefficient of the 4th term in the expansion of \((a + b)^{12}\)?
**Options:**
- 985
- 1,320
- 220
- 445
**Explanation:**
To solve for the coefficient of the 4th term in the binomial expansion of \((a+b)^{12}\), we use the binomial theorem formula:
\[
(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For the 4th term, we need the term where \(k=3\) (note that binomial theorem terms are indexed starting from \(k=0\)). The binomial coefficient \(\binom{12}{3}\) determines the coefficient we're looking for. We can compute it as follows:
\[
\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3! \cdot 9!}
\]
This simplifies to:
\[
\frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]
Therefore, the correct answer is:
- **220**
Feel free to use this derived explanation along with the question and provided options for educational purposes, helping students understand the binomial expansion and coefficient calculation methods.
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