Discussion
\(\frac{8}{4}({}^{7}P_{4})\)
Answer: D
\(\frac{8}{4}({}^{7}P_{4})\) equals \({}^{8}P_{4}\).
\[ \begin{aligned} {}^{7}P_{4} &= 7 \times 6 \times 5 \times 4 = 840 \\ \frac{8}{4} \times 840 &= 1680 = 8 \times 7 \times 6 \times 5 = {}^{8}P_{4} \end{aligned} \]
This is the identity \({}^{n}P_{r} = \frac{n}{n - r}\,{}^{n-1}P_{r}\) with \(n = 8\), \(r = 4\). For comparison, \({}^{8}P_{3} = 336\), \({}^{8}P_{5} = 6720\), \({}^{8}C_{3} = {}^{8}C_{5} = 56\).
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