Question

Show that if n is a positive integer with n ≥ 3, then

C(n, n − 2) = ((3n − 1)C(n, 3))/4

Answer 1

$$c(n, n-2)=2\dbinom{n}{3}+3\dbinom{n}{4}$$

$$=2\dfrac{n!}{3!(n-3)!}+3\dfrac{n!}{4!(n-4)!}$$

$$=\dfrac{n(n-1)(n-2)}{3}+\dfrac{n(n-1)(n-2)(n-3)}{8}$$

$$=\dfrac{n^3}{3}-n^2+\dfrac{2n}{3}+\dfrac{n^4}{8}-\dfrac{3n^3}{4}+\dfrac{11n^2}{8}-\dfrac{3n}{4}$$

$$=\dfrac{n^4}{8}-\dfrac{5n^3}{12}+\dfrac{3n^2}{8}-\dfrac{n}{12}$$

$$\dfrac{(3n-1)C(n,3)}{4}=\dfrac{(3n-1)n!}{3!(n-3)!(4)}$$

$$=\dfrac{(3n-1)(n)(n-1)(n-2)}{24}$$

$$=\dfrac{n^4}{8}-\dfrac{5n^3}{12}+\dfrac{3n^2}{8}-\dfrac{n}{12}$$

Hence

$$c(n, n-2)=\dfrac{(3n-1)C(n,3)}{4}$$