Simplify \(^{n + 1}C_{4} – ^{n – 1}C_{4}\)
= \(\frac{(n + 1)!}{4! (n – 3)!} – \frac{(n – 1)!}{4! (n – 5)!}\)
= \(\frac{(n + 1)(n)(n – 1)(n – 2)(n – 3)!}{4! (n – 3)!} – \frac{(n – 1)(n – 2)(n – 3)(n – 4)(n – 5)!}{4! (n – 5)!}\)
= \(\frac{(n + 1)(n)(n – 1)(n – 2)}{4!} – \frac{(n – 1)(n – 2)(n – 3)(n – 4)}{4!}\)
= \(\frac{(n – 1)(n – 2) [n(n + 1) – (n – 3)(n – 4)]}{4!}\)
= \(\frac{(n – 1)(n – 2) [n^{2} + n – n^{2} + 7n – 12]}{24}\)
= \(\frac{(n – 1)(n – 2)[8n – 12]}{24}\)
= \(\frac{(n – 1)(n – 2)(2n – 3)}{6}\)