To determine how many terms are in the binomial expansion of [tex]\((a + b)^8\)[/tex], we start by considering the general formula for the binomial expansion. The binomial expansion of [tex]\((a + b)^n\)[/tex] takes the form:
[tex]\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \][/tex]
This means that for any positive integer [tex]\(n\)[/tex], the expansion will include terms ranging from [tex]\(k = 0\)[/tex] to [tex]\(k = n\)[/tex]. Each value of [tex]\(k\)[/tex] corresponds to one unique term in the expansion.
To count the total number of terms present in this expansion, we note that [tex]\(k\)[/tex] covers all integers from 0 to [tex]\(n\)[/tex]. This implies that there are a total of [tex]\(n + 1\)[/tex] terms.
Given that [tex]\(n = 8\)[/tex] in the expansion [tex]\((a + b)^8\)[/tex]:
The number of terms is:
[tex]\[ 8 + 1 = 9 \][/tex]
Therefore, the binomial expansion of [tex]\((a + b)^8\)[/tex] includes [tex]\(\boxed{9}\)[/tex] terms.
