College Algebra, Chapter 9, 9.6, Section 9.6, Problem 34

Determine the first fifth terms in the expansion $(ab-1)^{20}$
Recall that the Binomial Theorem is defined as
Substituting $a = ab$ and $b = -1$ gives

$(ab - 1)^{20} = \left( \begin{array}{c} 20\\ 0 \end{array} \right) (ab)^{20} + \left( \begin{array}{c} 20\\ 1 \end{array} \right) (ab)^{19} (-1) + \left( \begin{array}{c} 20\\ 2 \end{array} \right) (ab)^{18} (-1)^2 + \left( \begin{array}{c} 20\\ 3 \end{array} \right) (ab)^{17} (-1)^3 + \left( \begin{array}{c} 20\\ 4 \end{array} \right) (ab)^{16} (-1)^4 + ....$

Thus, the 5th term is

$\begin{equation} \begin{aligned} &= \left( \begin{array}{c} 20\\ 4 \end{array} \right) (ab)^{16}(-1)^4\\ \\ &= \left( \frac{20!}{4!(20-4)!} \right) (ab)^{16}(-1)^4\\ \\ &= 4845 a^{16} b^{16} \end{aligned} \end{equation}$