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Notice again that \(P(10,4)\) starts out looking like \(10!\text{,}\) but we stop after 7. We can formally account for this “stopping” by dividing away the part of the factorial we do not want:
\begin{equation*}
P(10,4) = \frac{10\cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{10!}{6!}\text{.}
\end{equation*}
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