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! ...

Simplify \(^{n + 1}C_{4} - ^{n - 1}C_{4}\)

= \(\frac{(n + 1)!}{4! (n - 3)!} - \frac{(n - 1)!}{4! (n - 5)!}\)

= \(\frac{(n + 1)(n)(n - 1)(n - 2)(n - 3)!}{4! (n - 3)!} - \frac{(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)!}{4! (n - 5)!}\)

= \(\frac{(n + 1)(n)(n - 1)(n - 2)}{4!} - \frac{(n - 1)(n - 2)(n - 3)(n - 4)}{4!}\)

= \(\frac{(n - 1)(n - 2) [n(n + 1) - (n - 3)(n - 4)]}{4!}\)

= \(\frac{(n - 1)(n - 2) [n^{2} + n - n^{2} + 7n - 12]}{24}\)

= \(\frac{(n - 1)(n - 2)[8n -  12]}{24}\)

= \(\frac{(n - 1)(n - 2)(2n - 3)}{6}\)