\(^{n}P_{2} = \frac{n!}{(n - 2)!} = 20 \)

\(\frac{n(n - 1)(n - 2)!}{(n - 2)!} = 20\)

\(n(n - 1) = 20 \implies n^{2} - n - 20 = 0\)

\(n^{2} - 5n + 4n - 20 = 0\)

\(n(n - 5) + 4(n - 5) = 0\)

\(n = \text{5 or -4}\)

\(n = 5\)