Question

${}^n{C}_3{+}^n{C}_4{>}^{n+1}{C}_3$ Find the range of $n$

Answer 1

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

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

$\frac{(n+1)!(n+1-4)!}{4!}>\frac{(n+1)!(n+1-3)!}{3!}$

$\frac{(n-3)!}{4}>\frac{(n-2)!}{1}$

$\frac{(n-3)}{4}>1$

$\frac{n-3-4}{4}>0\Rightarrow \frac{n-7}{4}>0$

$nϵ(7,\infty )$