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}
$