## Comparing Binomial Terms Example

For what values of
$n$
is
${}^nC_3 - {}^{2n}C_2 \gt 0$
?
${}^nC_3= \frac{n!}{3!(n-3)!}=\frac{n(n-1)(n-2)}{6}$

${}^{2n}C_2= \frac{(2n)!}{2!(2n-2)!}=\frac{2n(2n-1)}{2}$

Then
$\frac{n(n-1)(n-2)}{6} - \frac{2n(2n-1)}{2} \gt 0$

$\frac{n}{6}((n-1)(n-2)-6(2n-1)) \gt 0$

$\frac{n}{6}(n^2-3n+2-12n+6) \gt 0$

$\frac{n}{6}(n^2-15n+8) \gt 0$

The quadratic inside the brackets is positive for
$n \ge 15$
so (1) is true for
$n \ge 15$
.