Answer
$n! \gt 2^n $ is true for $n \geq 4$
Work Step by Step
We need to prove that $n! \gt 2^n $ for $n \geq 4$
1. Our aim is to find that $P(n)$ is true for $n=4$
$4! \gt 2^4 \implies 24 \gt 8$
So, it is true for $n=4$.
2. Our aim is to find that $P(n)$ is true for $n=k$.This, it will also true for $n=k+1$
$(k+1) ! \gt 2^{k+1} \implies (k+1)(2^k) \gt 2^k (2) $
This yields: $ k+1 \gt 2$
So, it is true for $n=k+1$.
Therefore, $n! \gt 2^n $ is true for $n \geq 4$
