If \( n \ge 2 \) is a positive integer, then the sum of the series \( {}^{n+1}C_{2}+2({}^{2}C_{2}+{}^{3}C_{2}+{}^{4}C_{2}+\cdots+{}^{n}C_{2}) \) is:
\( \dfrac{n(n+1)^{2}(n+2)}{12} \)
\( \dfrac{n(n+1)(2n+1)}{6} \)
\( \dfrac{n(n-1)(2n+1)}{6} \)
\( \dfrac{n(2n+1)(3n+1)}{6} \)
