What Is The Coefficient Of The Third Term In The Binomial Expansion Of { (a+b)^6$}$?A. 1 B. 15 C. 20 D. 90

What is the Coefficient of the Third Term in the Binomial Expansion of (a+b)^6?

The binomial expansion is a mathematical concept used to expand expressions of the form (a+b)^n, where 'a' and 'b' are constants and 'n' is a positive integer. The binomial expansion is a crucial concept in algebra and is used to solve various mathematical problems. In this article, we will discuss the binomial expansion of (a+b)^6 and find the coefficient of the third term.

Understanding the Binomial Expansion

The binomial expansion of (a+b)^n is given by the formula:

(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 calculated as:

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

The binomial expansion is a sum of terms, where each term is a product of 'a' and 'b' raised to certain powers. The powers of 'a' and 'b' in each term are determined by the binomial coefficient.

Binomial Expansion of (a+b)^6

To find the binomial expansion of (a+b)^6, we will use the formula:

(a+b)^6 = {\sum_{k=0}^{{6}\binom{6}{k}a}{6-k}b^{k}$}$

We will calculate the binomial coefficients for each term and find the coefficient of the third term.

Calculating the Binomial Coefficients

The binomial coefficients for each term are calculated as:

{\binom{6}{0}=\frac{6!}{0!(6-0)!}=1$}$${\binom{6}{1}=\frac{6!}{1!(6-1)!}=6\$} {\binom{6}{2}=\frac{6!}{2!(6-2)!}=15$}$${\binom{6}{3}=\frac{6!}{3!(6-3)!}=20\$} {\binom{6}{4}=\frac{6!}{4!(6-4)!}=15$}$${\binom{6}{5}=\frac{6!}{5!(6-5)!}=6\$} {\binom{6}{6}=\frac{6!}{6!(6-6)!}=1$}$

Finding the Coefficient of the Third Term

The third term in the binomial expansion of (a+b)^6 is {a^{6-3}b{3}=a^{3}b{3}$}$. The coefficient of this term is the binomial coefficient {\binom{6}{3}$}$, which is 20.

Conclusion

In this article, we discussed the binomial expansion of (a+b)^6 and found the coefficient of the third term. The binomial expansion is a crucial concept in algebra and is used to solve various mathematical problems. The coefficient of the third term in the binomial expansion of (a+b)^6 is 20.

Frequently Asked Questions

• What is the binomial expansion of (a+b)^6?
• How do you calculate the binomial coefficients?
• What is the coefficient of the third term in the binomial expansion of (a+b)^6?

Answers

• The binomial expansion of (a+b)^6 is {\sum_{k=0}^{{6}\binom{6}{k}a}{6-k}b^{k}$}$.
• The binomial coefficients are calculated as {\binom{n}{k}=\frac{n!}{k!(n-k)!}$}$.
• The coefficient of the third term in the binomial expansion of (a+b)^6 is 20.

References

• "Binomial Expansion" by Math Open Reference
• "Binomial Coefficient" by Wolfram MathWorld
• "Binomial Theorem" by Khan Academy
  Binomial Expansion Q&A

In this article, we will answer some frequently asked questions about the binomial expansion.

Q: What is the binomial expansion?

A: The binomial expansion is a mathematical concept used to expand expressions of the form (a+b)^n, where 'a' and 'b' are constants and 'n' is a positive integer.

Q: How do you calculate the binomial coefficients?

A: The binomial coefficients are calculated using the formula:

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

where n is the exponent and k is the term number.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical concept that describes the expansion of expressions of the form (a+b)^n. It is a generalization of the binomial expansion and is used to expand expressions of the form (a+b)^n, where n is a positive integer.

Q: How do you use the binomial theorem?

A: The binomial theorem is used to expand expressions of the form (a+b)^n, where n is a positive integer. It is a powerful tool for solving mathematical problems and is used in many areas of mathematics, including algebra, geometry, and calculus.

Q: What is the difference between the binomial expansion and the binomial theorem?

A: The binomial expansion is a specific case of the binomial theorem, where n is a positive integer. The binomial theorem is a more general concept that describes the expansion of expressions of the form (a+b)^n, where n is a positive integer.

Q: How do you find the coefficient of a term in the binomial expansion?

A: The coefficient of a term in the binomial expansion is found by multiplying the binomial coefficient by the powers of 'a' and 'b' in the term.

Q: What is the significance of the binomial expansion?

A: The binomial expansion is a fundamental concept in mathematics and has many applications in science, engineering, and economics. It is used to solve problems in probability, statistics, and combinatorics, and is a key tool for understanding many mathematical concepts.

Q: Can you give an example of how to use the binomial expansion?

A: Yes, here is an example of how to use the binomial expansion to expand the expression (a+b)^6:

(a+b)^6 = {\sum_{k=0}^{{6}\binom{6}{k}a}{6-k}b^{k}$}$

Using the binomial coefficients, we can expand this expression as:

(a+b)^6 = {a^{6}+6a{5}b+15a^{4}b{2}+20a^{3}b{3}+15a^{2}b{4}+6ab^{5}+b{6}$}$

Q: What are some common applications of the binomial expansion?

A: The binomial expansion has many applications in science, engineering, and economics. Some common applications include:

• Probability and statistics: The binomial expansion is used to calculate probabilities and expected values in probability theory.
• Combinatorics: The binomial expansion is used to count the number of ways to arrange objects in combinatorics.
• Algebra: The binomial expansion is used to solve equations and inequalities in algebra.
• Geometry: The binomial expansion is used to calculate areas and volumes in geometry.
• Calculus: The binomial expansion is used to calculate derivatives and integrals in calculus.

Q: Can you give some tips for using the binomial expansion?

A: Yes, here are some tips for using the binomial expansion:

• Make sure to use the correct formula for the binomial coefficients.
• Use the binomial expansion to expand expressions of the form (a+b)^n, where n is a positive integer.
• Use the binomial expansion to solve problems in probability, statistics, and combinatorics.
• Use the binomial expansion to calculate areas and volumes in geometry.
• Use the binomial expansion to calculate derivatives and integrals in calculus.

Q: What are some common mistakes to avoid when using the binomial expansion?

A: Some common mistakes to avoid when using the binomial expansion include:

• Using the wrong formula for the binomial coefficients.
• Failing to use the correct powers of 'a' and 'b' in the terms.
• Failing to use the correct binomial coefficients for the terms.
• Failing to simplify the expression after expanding it.

Q: Can you give some examples of how to use the binomial expansion in real-world problems?

A: Yes, here are some examples of how to use the binomial expansion in real-world problems:

• A company is producing a new product and wants to calculate the probability of success. The company uses the binomial expansion to calculate the probability of success.
• A scientist is studying the behavior of a population and wants to calculate the expected value of a variable. The scientist uses the binomial expansion to calculate the expected value.
• An engineer is designing a new system and wants to calculate the area of a shape. The engineer uses the binomial expansion to calculate the area.

Q: What are some advanced topics related to the binomial expansion?

A: Some advanced topics related to the binomial expansion include:

• The multinomial theorem: This is a generalization of the binomial theorem that describes the expansion of expressions of the form (a+b+c+...)^n, where n is a positive integer.
• The binomial distribution: This is a probability distribution that is used to model the number of successes in a fixed number of independent trials.
• The Poisson distribution: This is a probability distribution that is used to model the number of events in a fixed interval of time or space.

Q: Can you give some resources for learning more about the binomial expansion?

A: Yes, here are some resources for learning more about the binomial expansion:

• "Binomial Expansion" by Math Open Reference
• "Binomial Coefficient" by Wolfram MathWorld
• "Binomial Theorem" by Khan Academy
• "Binomial Distribution" by Wikipedia
• "Poisson Distribution" by Wikipedia