Each Answer Choice Below Shows The Entries In A Row Of Pascal's Triangle. Which Lists The Coefficients In The Binomial Expansion Of \[$(a+b)^8\$\]?A. \[$1, 7, 21, 35, 21, 7, 1\$\]B. \[$1, 7, 21, 35, 35, 21, 7, 1\$\]C. \[$1,

Introduction

Pascal's triangle is a mathematical concept that has been widely used in various fields, including algebra, geometry, and combinatorics. It is a triangular array of the binomial coefficients, where each number is the sum of the two numbers directly above it. The binomial expansion, on the other hand, is a mathematical concept that represents the expansion of a binomial expression raised to a power. In this article, we will explore the relationship between Pascal's triangle and the binomial expansion, and determine which list of coefficients corresponds to the binomial expansion of ${(a+b)^8}$.

Pascal's Triangle

Pascal's triangle is a triangular array of numbers that starts with a single 1 at the top, and each subsequent row is obtained by adding the two numbers directly above it. The first few rows of Pascal's triangle are:

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
 1 5 10 10 5 1

Binomial Expansion

The binomial expansion is a mathematical concept that represents the expansion of a binomial expression raised to a power. The binomial 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, which is given by:

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

Coefficients in the Binomial Expansion

The coefficients in the binomial expansion of ${(a+b)^8}$ can be obtained by using the formula for the binomial coefficient. The first few coefficients are:

${\binom{8}{0} = 1}$ ${\binom{8}{1} = 8}$ ${\binom{8}{2} = 28}$ ${\binom{8}{3} = 56}$ ${\binom{8}{4} = 70}$ ${\binom{8}{5} = 56}$ ${\binom{8}{6} = 28}$ ${\binom{8}{7} = 8}$ ${\binom{8}{8} = 1}$

Comparing the Coefficients

Now that we have obtained the coefficients in the binomial expansion of ${(a+b)^8}$, we can compare them with the given answer choices. The correct list of coefficients should match the coefficients we obtained using the formula for the binomial coefficient.

Answer Choice A

The first answer choice is:

${1, 7, 21, 35, 21, 7, 1}$

This list of coefficients does not match the coefficients we obtained using the formula for the binomial coefficient.

Answer Choice B

The second answer choice is:

${1, 7, 21, 35, 35, 21, 7, 1}$

This list of coefficients does not match the coefficients we obtained using the formula for the binomial coefficient.

Answer Choice C

The third answer choice is:

${1, 8, 28, 56, 70, 56, 28, 8, 1}$

This list of coefficients does not match the coefficients we obtained using the formula for the binomial coefficient.

Conclusion

Based on our analysis, we can conclude that none of the given answer choices match the coefficients in the binomial expansion of ${(a+b)^8}$. However, if we look at the coefficients we obtained using the formula for the binomial coefficient, we can see that the correct list of coefficients is:

${1, 8, 28, 56, 70, 56, 28, 8, 1}$

This list of coefficients is not among the given answer choices. Therefore, we cannot determine which answer choice is correct based on the information provided.

Final Answer

Unfortunately, we cannot determine which answer choice is correct based on the information provided. However, we can conclude that the correct list of coefficients in the binomial expansion of ${(a+b)^8}$ is:

${1, 8, 28, 56, 70, 56, 28, 8, 1}$

This list of coefficients can be obtained using the formula for the binomial coefficient.