- Apply the binomial theorem formula \((A+B)^n = \sum_{k=0}^{n} \binom{n}{k} A^{n-k} B^k\) with \(A=2x\), \(B=-3y\), and \(n=10\).
- Determine that the \(x^5y^5\) term occurs when \(k=5\).
- The coefficient is \(\binom{10}{5} (2)^5 (-3)^5\).
- The correct option is d. \(\boxed{d}\)
Explanation
- Using the Binomial Theorem
To find the coefficient of the \(x^5y^5\) term in the expansion of \((2x - 3y)^{10}\), we utilize the Binomial Theorem. The Binomial Theorem states that for any non-negative integer \(n\), 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 our case, we have \(A = 2x\), \(B = -3y\), and \(n = 10\). The expression becomes \((2x + (-3y))^{10}\). The general term in this expansion is:
\[T_{k+1} = \binom{10}{k} (2x)^{10-k} (-3y)^k\]
- Identifying the Specific Term
We want to isolate the term containing \(x^5y^5\). Looking at our general term formula:
\[T_{k+1} = \binom{10}{k} (2x)^{10-k} (-3y)^k = \binom{10}{k} 2^{10-k} x^{10-k} (-3)^k y^k\]
To have the powers of \(x\) and \(y\) both equal to 5, we must set the exponent of \(x\) (which is \(10-k\)) equal to 5, and the exponent of \(y\) (which is \(k\)) equal to 5.
Solving \(10-k = 5\) gives \(k = 5\).
Substituting \(k = 5\) into the formula, we find the relevant term:
\[T_{5+1} = \binom{10}{5} 2^{10-5} x^5 (-3)^5 y^5 = \binom{10}{5} 2^5 (-3)^5 x^5 y^5\]
- Identifying the Correct Coefficient
Now we identify the coefficient by extracting the numerical part from the expression \(\binom{10}{5} 2^5 (-3)^5 x^5 y^5\).
The coefficient is \(\binom{10}{5} 2^5 (-3)^5\).
Comparing this to the provided options:
A\({ }_{10} C_5(2)^5(3)^5\)
B\(-{ }_{10} C_5(2)^5(-3)^5\)
C\({ }_{10} C_5(2)^5(3)\)
D\({ }_{10} C_5(2)^5(-3)^5\)
Option d matches our derived coefficient exactly.
Examples
Binomial expansions are used in various fields, such as probability and finance. For instance, if you are calculating the likelihood of a specific outcome in a series of independent events (like a sequence of 10 coin flips or business transactions), the binomial theorem helps determine the weights associated with each specific outcome combination. By expanding expressions like \((ax + by)^n\), analysts can easily find the coefficient of a specific state to assess its frequency or impact in a complex system.