What is the ninth term in the binomial expansion of $(x-2 y)^{13}$?
A. $329,472 x^5 y^8$
B. $-329,472 x^5 y^8$
C. $-41,184 x^8 y^5$
D. $41,184 x^8 y^5$
Answer (2)
The ninth term in the binomial expansion of ( x − 2 y ) 13 is 329472 x 5 y 8 . This is obtained by calculating the binomial coefficient and substituting the appropriate values into the binomial expansion formula. The confirmed answer is option A: 329472 x 5 y 8 .
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Identify that the binomial expansion of ( x − 2 y ) 13 is needed.
Determine that the ninth term corresponds to k = 8 in the binomial coefficient.
Calculate the binomial coefficient ( 8 13 ) = 1287 .
Compute the ninth term as 1287 ⋅ x 5 ⋅ ( − 2 y ) 8 = 329472 x 5 y 8 , so the final answer is 329 , 472 x 5 y 8 .
Explanation
Understanding the Binomial Theorem We are asked to find the ninth term in the binomial expansion of ( x − 2 y ) 13 . Let's recall the binomial theorem, which provides a way to expand expressions of the form ( a + b ) n . The general term in the binomial expansion is given by the formula: ( k n ) a n − k b k where ( k n ) is the binomial coefficient, also written as C ( n , k ) or n C k , which represents the number of ways to choose k elements from a set of n elements.
Identifying the Values In our case, we have a = x , b = − 2 y , and n = 13 . We want to find the ninth term, which corresponds to k = 8 (since the first term corresponds to k = 0 ). So, we need to calculate the term with k = 8 in the expansion of ( x − 2 y ) 13 .
Calculating the Binomial Coefficient The ninth term is given by: ( 8 13 ) x 13 − 8 ( − 2 y ) 8 Let's break this down into smaller parts. First, we need to calculate the binomial coefficient ( 8 13 ) , which is: ( 8 13 ) = 8 ! ( 13 − 8 )! 13 ! = 8 ! 5 ! 13 ! = 5 × 4 × 3 × 2 × 1 13 × 12 × 11 × 10 × 9 = 1287
Calculating the Powers Next, we calculate x 13 − 8 = x 5 . Then, we calculate ( − 2 y ) 8 = ( − 2 ) 8 y 8 = 256 y 8 .
Finding the Ninth Term Now, we multiply all these parts together: ( 8 13 ) x 13 − 8 ( − 2 y ) 8 = 1287 x 5 ( 256 y 8 ) = 329472 x 5 y 8 Therefore, the ninth term in the binomial expansion of ( x − 2 y ) 13 is 329472 x 5 y 8 .
Final Answer Thus, the ninth term in the binomial expansion of ( x − 2 y ) 13 is 329472 x 5 y 8 .
Examples
Binomial expansion is used in probability calculations, such as determining the likelihood of a certain number of successes in a series of independent trials. For example, if you flip a coin 13 times, the binomial expansion can help you calculate the probability of getting exactly 8 heads. It also has applications in physics, statistics, computer science, and finance, where understanding combinations and probabilities is crucial. In finance, it can be used to model the distribution of returns on an investment.
