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Section 21.1 Factorials
In counting, factorials come up a lot.
\(n!\)
for natural number \(n\text{,}\) notation for the computation formula
\begin{equation*}
n (n - 1) (n - 2) \dotsm 2 \cdot 1
\end{equation*}
Example 21.1.1 . Two factorial calculations.
\begin{align*}
3! \amp = 3 \cdot 2 \cdot 1 = 6, \amp
7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5,040.
\end{align*}
Example 21.1.2 . Factorial factors.
A factorial contains every smaller factorial as a factor. For example,
\begin{equation*}
\frac{7!}{3!}
= \frac{
7 \cdot 6 \cdot 5 \cdot 4 \cdot \cancel{(3!)}
}{
\cancel{3!}
}
= 7 \cdot 6 \cdot 5 \cdot 4 = 840\text{.}
\end{equation*}
