What Is The Coefficient Of The $x^6 Y^3$ Term In The Expansion Of $(x+2y)^9$?A. 84 B. 168 C. 336 D. 672

by ADMIN 111 views

Introduction

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. The theorem states that the expansion of $(a+b)^n$ is given by:

(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

where $\binom{n}{k}$ is the binomial coefficient, which is defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

In this article, we will use the binomial theorem to find the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$.

The Binomial Theorem

The binomial theorem is a fundamental concept in algebra that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. The theorem states that the expansion of $(a+b)^n$ is given by:

(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

where $\binom{n}{k}$ is the binomial coefficient, which is defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

The binomial theorem can be used to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. The theorem is a powerful tool in algebra that allows us to simplify complex expressions and solve problems in a variety of fields, including mathematics, physics, and engineering.

Finding the Coefficient of the $x^6 y^3$ Term

To find the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$, we can use the binomial theorem. The binomial theorem states that the expansion of $(a+b)^n$ is given by:

(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

In this case, we have $(x+2y)^9$, so we can substitute $a=x$, $b=2y$, and $n=9$ into the binomial theorem:

(x+2y)9=∑k=09(9k)x9−k(2y)k(x+2y)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (2y)^k

To find the coefficient of the $x^6 y^3$ term, we need to find the term in the expansion where $x$ has a power of $6$ and $y$ has a power of $3$. This means that we need to find the term where $k=3$, since $k$ represents the power of $y$.

Calculating the Coefficient

To calculate the coefficient of the $x^6 y^3$ term, we need to substitute $k=3$ into the binomial theorem:

(x+2y)9=∑k=09(9k)x9−k(2y)k(x+2y)^9 = \sum_{k=0}^{9} \binom{9}{k} x^{9-k} (2y)^k

=(93)x9−3(2y)3= \binom{9}{3} x^{9-3} (2y)^3

=(93)x6(2y)3= \binom{9}{3} x^6 (2y)^3

=(93)x68y3= \binom{9}{3} x^6 8y^3

=84x68y3= 84 x^6 8y^3

=672x6y3= 672 x^6 y^3

Therefore, the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$ is $672$.

Conclusion

In this article, we used the binomial theorem to find the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$. We substituted $a=x$, $b=2y$, and $n=9$ into the binomial theorem and found the term where $x$ has a power of $6$ and $y$ has a power of $3$. We then calculated the coefficient of this term and found that it is $672$. This result can be used to solve problems in a variety of fields, including mathematics, physics, and engineering.

Final Answer

The final answer is: 672\boxed{672}

Introduction

In our previous article, we used the binomial theorem to find the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$. In this article, we will answer some common questions related to the binomial theorem and the coefficient of the $x^6 y^3$ term.

Q1: What is the binomial theorem?

A1: The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer. The theorem states that the expansion of $(a+b)^n$ is given by:

(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Q2: How do I use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$?

A2: To use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$, you need to follow these steps:

  1. Identify the term you want to find the coefficient of.
  2. Determine the power of $a$ and $b$ in the term.
  3. Substitute the values of $a$, $b$, and $n$ into the binomial theorem.
  4. Simplify the expression to find the coefficient of the term.

Q3: What is the coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$?

A3: The coefficient of the $x^6 y^3$ term in the expansion of $(x+2y)^9$ is $672$.

Q4: How do I calculate the binomial coefficient $\binom{n}{k}$?

A4: The binomial coefficient $\binom{n}{k}$ is calculated using the formula:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Q5: What is the significance of the binomial theorem in mathematics and other fields?

A5: The binomial theorem is a fundamental concept in mathematics that has numerous applications in other fields, including physics, engineering, and computer science. It is used to solve problems involving combinations, permutations, and probability.

Q6: Can I use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$ where $a$ and $b$ are complex numbers?

A6: Yes, you can use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$ where $a$ and $b$ are complex numbers. However, you need to be careful when simplifying the expression, as complex numbers can lead to complex coefficients.

Q7: How do I apply the binomial theorem to solve problems in physics and engineering?

A7: The binomial theorem can be used to solve problems involving combinations, permutations, and probability in physics and engineering. For example, it can be used to calculate the probability of a particle decaying into a specific state, or to determine the number of possible arrangements of a system.

Q8: Can I use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$ where $n$ is a negative integer?

A8: No, you cannot use the binomial theorem to find the coefficient of a term in the expansion of $(a+b)^n$ where $n$ is a negative integer. The binomial theorem only applies to positive integers.

Q9: How do I simplify the expression for the coefficient of a term in the expansion of $(a+b)^n$?

A9: To simplify the expression for the coefficient of a term in the expansion of $(a+b)^n$, you need to follow these steps:

  1. Simplify the binomial coefficient $\binom{n}{k}$.
  2. Simplify the powers of $a$ and $b$.
  3. Combine like terms.

Q10: What are some common applications of the binomial theorem in mathematics and other fields?

A10: The binomial theorem has numerous applications in mathematics and other fields, including:

  • Calculating combinations and permutations
  • Determining the number of possible arrangements of a system
  • Calculating probabilities
  • Solving problems involving complex numbers
  • Calculating the coefficient of a term in the expansion of $(a+b)^n$

Conclusion

In this article, we answered some common questions related to the binomial theorem and the coefficient of the $x^6 y^3$ term. We hope that this article has provided you with a better understanding of the binomial theorem and its applications in mathematics and other fields.