To determine which expression represents the sixth term in the binomial expansion of [tex]\((2a - 3b)^{10}\)[/tex], we can follow these steps:
1. Identify the general term in a binomial expansion: The general term in the expansion of [tex]\((x + y)^n\)[/tex] is given by:
[tex]\[
T_{k+1} = \binom{n}{k} x^{n-k} y^k
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient.
2. Apply the general term to the given binomial expression: For [tex]\((2a - 3b)^{10}\)[/tex], we can let [tex]\(x = 2a\)[/tex] and [tex]\(y = -3b\)[/tex]. The general term becomes:
[tex]\[
T_{k+1} = \binom{10}{k} (2a)^{10-k} (-3b)^k
\][/tex]
3. Identify the specific term we need: The question asks for the sixth term in the expansion. Remember that terms in the binomial expansion are 1-indexed, meaning the first term corresponds to [tex]\(k = 0\)[/tex], the second term corresponds to [tex]\(k = 1\)[/tex], and so forth. Thus, the sixth term corresponds to [tex]\(k = 5\)[/tex].
Plug [tex]\(k = 5\)[/tex] into the general term expression:
[tex]\[
T_{6} = \binom{10}{5} (2a)^{10-5} (-3b)^5
\][/tex]
4. Simplify the expression: Substitute [tex]\(10 - 5 = 5\)[/tex] into the powers:
[tex]\[
T_{6} = \binom{10}{5} (2a)^5 (-3b)^5
\][/tex]
The simplified expression matching one of the choices given is:
[tex]\[
\boxed{{ }_{10}C_5 (2a)^5 (-3b)^5}
\][/tex]
