What is the coefficient of æ10 in (æ + 3)13?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discrete Math: Binomial Coefficients
![### Problem Statement
**Question:**
What is the coefficient of \( x^{10} \) in \( (x + 3)^{13} \)?
[Answer box provided for input]
### Explanation:
To find the coefficient of \( x^{10} \) in the expansion of \( (x + 3)^{13} \), we use the Binomial Theorem. The Binomial Theorem states that:
\[
(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k
\]
In this case, \( n = 13 \), \( a = 3 \), and we need the term where the exponent of \( x \) is 10. Thus, we need the term where \( n-k = 10 \), which implies \( k = 3 \).
The coefficient of \( x^{10} \) is given by:
\[
\binom{13}{3} \cdot 3^3
\]
Calculating:
1. **Binomial Coefficient:**
\[
\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286
\]
2. **Calculating \( 3^3 \):**
\[
3^3 = 27
\]
3. **Coefficient Calculation:**
\[
286 \times 27 = 7722
\]
Therefore, the coefficient of \( x^{10} \) is 7722.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76b39ad3-98e6-4bd8-afd7-9890d91d962f%2Fdec0f3d0-76be-44be-9076-d922fb8c2d4f%2F5essq5_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question:**
What is the coefficient of \( x^{10} \) in \( (x + 3)^{13} \)?
[Answer box provided for input]
### Explanation:
To find the coefficient of \( x^{10} \) in the expansion of \( (x + 3)^{13} \), we use the Binomial Theorem. The Binomial Theorem states that:
\[
(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k
\]
In this case, \( n = 13 \), \( a = 3 \), and we need the term where the exponent of \( x \) is 10. Thus, we need the term where \( n-k = 10 \), which implies \( k = 3 \).
The coefficient of \( x^{10} \) is given by:
\[
\binom{13}{3} \cdot 3^3
\]
Calculating:
1. **Binomial Coefficient:**
\[
\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286
\]
2. **Calculating \( 3^3 \):**
\[
3^3 = 27
\]
3. **Coefficient Calculation:**
\[
286 \times 27 = 7722
\]
Therefore, the coefficient of \( x^{10} \) is 7722.
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