To find the coefficient of the sixth term in the binomial expansion of [tex]\((a+b)^9\)[/tex], we can use the Binomial Theorem.
The Binomial Theorem states that:
[tex]\[
(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, also known as "n choose k," which can be calculated using the formula:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
In this case, the exponent [tex]\(n\)[/tex] is 9. To find the coefficient of the sixth term, we need to determine the corresponding term in the expansion. The general term in the binomial expansion is given by:
[tex]\[
\binom{n}{k} a^{n-k} b^k
\][/tex]
Note that the terms are counted starting from [tex]\(k=0\)[/tex]. So, the sixth term corresponds to [tex]\(k=5\)[/tex]. Therefore, we need to find the binomial coefficient [tex]\(\binom{9}{5}\)[/tex].
The binomial coefficient [tex]\(\binom{9}{5}\)[/tex] can be calculated as follows:
[tex]\[
\binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}
\][/tex]
Using the pre-determined numerical result, we have that:
[tex]\[
\binom{9}{5} = 126
\][/tex]
Therefore, the coefficient of the sixth term in the binomial expansion of [tex]\((a+b)^9\)[/tex] is:
[tex]\[
\boxed{126}
\][/tex]
