- Identify the binomial parameters: \(x=2a\), \(y=-3b\), and \(n=10\).
- Apply the general term formula for the binomial expansion, where the \(k\)-th term is \(T_{k+1} = \binom{n}{k} x^{n-k} y^k\).
- For the sixth term (\(k=5\)), substitute the parameters to get \({}_{10}C_5(2a)^5(-3b)^5\).
- Match the result with the given choices to find that the correct expression is \(\boxed{b}\).
Explanation
- Understanding the Binomial Theorem and Identifying Parameters
To find the sixth term of the expansion of \((2a - 3b)^{10}\), we use the Binomial Theorem. The Binomial Theorem states that for any non-negative integer \(n\), the expansion of \((x + y)^n\) is given by:
\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
In our specific case, \(x = 2a\), \(y = -3b\), and \(n = 10\). The terms in this expansion are indexed by \(k\), starting from \(k = 0\) for the first term, \(k = 1\) for the second term, and so on. Therefore, the \(m\)-th term corresponds to \(k = m - 1\).
- Calculating the Sixth Term
We are looking for the sixth term, so we set \(m = 6\), which means \(k = 6 - 1 = 5\). The general formula for the \((k+1)\)-th term is:
\[T_{k+1} = \binom{n}{k} x^{n-k} y^k\]
Substituting our values \(n = 10\), \(k = 5\), \(x = 2a\), and \(y = -3b\), we get:
\[T_{6} = \binom{10}{5} (2a)^{10-5} (-3b)^5\]
\[T_{6} = \binom{10}{5} (2a)^5 (-3b)^5\]
Note that \(\binom{10}{5}\) is the same as \({}_{10}C_5\), which is a common notation for binomial coefficients. Thus, the expression is \({}_{10}C_5(2a)^5(-3b)^5\).
- Matching with the Given Options
Comparing our result \({}_{10}C_5(2a)^5(-3b)^5\) with the given options, we can see that it matches option 'b'. The process shows that the expansion term is specifically determined by the values of the coefficients and the exponents assigned to each variable component. The final answer is option 'b'.
Examples
Binomial expansion is widely used in probability theory and statistics, particularly when dealing with sequences of independent experiments like flipping a coin or calculating the likelihood of certain outcomes in a trial. For example, if you are calculating the probability of getting exactly 6 heads in 10 tosses of a biased coin, the binomial theorem provides the exact term needed to account for all combinations of outcomes. Understanding this algebraic structure allows for precise modeling of risk and expected outcomes in fields ranging from finance to manufacturing quality control.