Evaluating the Binomial Coefficient

Oct 13, 2025 by miodeya.com 107 views

Introduction

The binomial theorem is a powerful tool in mathematics 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 = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n

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

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

In this article, we will use the binomial theorem to find the value of the 4th term of the expansion of $(y+2z)^8$ .

The Binomial Theorem

The binomial theorem is a fundamental concept in mathematics that has numerous applications in various fields, including algebra, geometry, and calculus. The theorem 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 binomial theorem can be stated as:

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

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

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

Expanding $(y+2z)^8$

To find the value of the 4th term of the expansion of $(y+2z)^8$ , we need to expand the expression using the binomial theorem. We can do this by substituting $a=y$ , $b=2z$ , and $n=8$ into the formula:

(y+2z)^8 = \sum_{k=0}^{8} \binom{8}{k} y^{8-k} (2z)^k

Finding the 4th Term

The 4th term of the expansion is given by the term where $k=3$ . We can find this term by substituting $k=3$ into the formula:

\binom{8}{3} y^{8-3} (2z)^3 = \binom{8}{3} y^5 (2z)^3

Evaluating the Binomial Coefficient

The binomial coefficient $\binom{8}{3}$ can be evaluated as:

\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8\cdot7\cdot6\cdot5!}{3\cdot2\cdot1\cdot5!} = \frac{8\cdot7\cdot6}{3\cdot2\cdot1} = 56

Evaluating the Term

Now that we have evaluated the binomial coefficient, we can evaluate the term:

\binom{8}{3} y^5 (2z)^3 = 56y^5 (2z)^3 = 56\cdot2^3y^5z^3 = 448y^5z^3

Conclusion

In this article, we used the binomial theorem to find the value of the 4th term of the expansion of $(y+2z)^8$ . We evaluated the binomial coefficient and the term, and found that the 4th term is equal to $448y^5z^3$ . This result is consistent with the answer choices provided.

Answer

The value of the 4th term of the expansion of $(y+2z)^8$ is $\boxed{448y^5z^3}$ .

Introduction

In our previous article, we used the binomial theorem to find the value of the 4th term of the expansion of $(y+2z)^8$ . In this article, we will answer some frequently asked questions related to the binomial theorem and the expansion of $(y+2z)^8$ .

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool in mathematics 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 = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n

Q: How do I use the binomial theorem to expand an expression?

A: To use the binomial theorem to expand an expression, you need to substitute the values of $a$ , $b$ , and $n$ into the formula. For example, to expand $(y+2z)^8$ , you would substitute $a=y$ , $b=2z$ , and $n=8$ into the formula:

(y+2z)^8 = \sum_{k=0}^{8} \binom{8}{k} y^{8-k} (2z)^k

Q: What is the 4th term of the expansion of $(y+2z)^8$ ?

A: The 4th term of the expansion of $(y+2z)^8$ is given by the term where $k=3$ . We can find this term by substituting $k=3$ into the formula:

\binom{8}{3} y^{8-3} (2z)^3 = \binom{8}{3} y^5 (2z)^3

Q: How do I evaluate the binomial coefficient?

A: The binomial coefficient $\binom{n}{k}$ can be evaluated as:

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

For example, to evaluate the binomial coefficient $\binom{8}{3}$ , you would calculate:

\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8\cdot7\cdot6\cdot5!}{3\cdot2\cdot1\cdot5!} = \frac{8\cdot7\cdot6}{3\cdot2\cdot1} = 56

Q: What is the value of the 4th term of the expansion of $(y+2z)^8$ ?

A: The value of the 4th term of the expansion of $(y+2z)^8$ is:

\binom{8}{3} y^5 (2z)^3 = 56y^5 (2z)^3 = 56\cdot2^3y^5z^3 = 448y^5z^3

Q: Can I use the binomial theorem to expand expressions with negative exponents?

A: Yes, you can use the binomial theorem to expand expressions with negative exponents. However, you need to be careful when dealing with negative exponents, as they can lead to complex expressions.

Q: Can I use the binomial theorem to expand expressions with fractional exponents?

A: Yes, you can use the binomial theorem to expand expressions with fractional exponents. However, you need to be careful when dealing with fractional exponents, as they can lead to complex expressions.

Conclusion

In this article, we answered some frequently asked questions related to the binomial theorem and the expansion of $(y+2z)^8$ . We hope that this article has been helpful in clarifying any doubts you may have had about the binomial theorem and its applications.