What is the term in the expansion of $(x-2 y)^{13}$?
The binomial theorem states that the expansion of \((a+b)^n\) is given by:
\[(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]
In this problem, we are looking for a term in the expansion of \((x - 2y)^{13}\). Here, \(a = x\), \(b = -2y\), and \(n = 13\). The general term in the expansion, denoted by \(T_{k+1}\), is given by:
\[T_{k+1} = \binom{13}{k} x^{13-k} (-2y)^k\]
This can be simplified by separating the variables and the constant coefficient:
\[T_{k+1} = \binom{13}{k} x^{13-k} (-2)^k y^k = \binom{13}{k} (-2)^k x^{13-k} y^k\]
To determine the potential terms, we test values of \(k\) that match the powers of \(x\) and \(y\) in the provided options.
The provided options are categorized by powers of \(x\) and \(y\). Specifically, we have terms involving \(x^5 y^8\) and \(x^8 y^5\).
Matching \(x^{13-k} y^k\) to \(x^5 y^8\) gives \(k = 8\).
Matching \(x^{13-k} y^k\) to \(x^8 y^5\) gives \(k = 5\).
First, for \(k = 8\):
The binomial coefficient \(\binom{13}{8} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1,287\).
The power of the coefficient \((-2)^8 = 256\).
The numerical coefficient is \(\binom{13}{8} \times (-2)^8 = 1,287 \times 256 = 329,472\).
Thus, the term is \(329,472 x^5 y^8\).
Second, for \(k = 5\):
The binomial coefficient \(\binom{13}{5} = \binom{13}{8} = 1,287\).
The power of the coefficient \((-2)^5 = -32\).
The numerical coefficient is \(\binom{13}{5} \times (-2)^5 = 1,287 \times -32 = -41,184\).
Thus, the term is \(-41,184 x^8 y^5\).
Comparing our results with the provided options:
Our calculations confirm that \(329,472 x^5 y^8\) (Option b) and \(-41,184 x^8 y^5\) (Option c) are valid terms in the expansion of \((x - 2y)^{13}\). Since the question asks for a term in the expansion, both options b and c are correct. However, checking typical question formats, usually one answer is expected. Given both are valid, let us specify the correct ones as \(\boxed{\text{b and c}}\).
The binomial theorem is used in fields like probability and statistics to determine the distribution of independent events. For instance, if you are modeling the probability of flipping a coin or the growth of a business strategy with several independent variables, binomial expansions allow you to calculate the probability or impact of specific combinations of outcomes efficiently. This is a fundamental tool for engineers, economists, and data analysts to predict outcomes in complex systems.