Question

Prove that ${}^{n-1}{C}_3{+}^{n-1}{C}_4{>}^n{C}_3$ if n > 7.

Answer 1

${}^{n-1}{C}_3{+}^{n-1}{C}_4{>}^n{C}_3$

We know that ${}^n{C}_r{+}^n{C}_{r+1}{=}^{n+1}{C}_{r+1}$

${}^{n-1}{C}_3{+}^{n-1}{C}_4{=}^n{C}_4$

${\Rightarrow }^n{C}_4{>}^n{C}_3$

$\Rightarrow \frac{n!}{4!(n-4)!}>\frac{n!}{3!(n-3)!}$

$\Rightarrow \frac{(n-3)!}{(n-4)!}>\frac{4!}{3!}$

$\Rightarrow n-3>4$

$\therefore n>4$

$\therefore n>7$

hence prooved