2 ^ n < n ! is true for \ wheren ∈ N\

\text{P}\left(\text{n}\right):{2}^{n} < n !$<br /><br />$\text{P} \left(\text{1}\right), \text{P} \left(2\right), \text{P}\left(3\right)$are not true.<br /><br />$\text{P} \left(\text{4}\right)$:16 < 24$ is true

\text{P}\left(\text{K}\right)$:$K ! > {2}^{K}$is true for some$K \in \mathbb{N}, K > 4$<br /><br />To prove$P ( K + 1 ) : \left(K + 1\right)$! > {2}^{K + 1}$ is true

Multiplying both sides of \text{P} \left(\text{K}\right)$by$( K + 1 ),$we get,<br /><br />\left(K + 1\right)$! > ${2}^{K}\left(K + 1\right) > {2}^{K}. 2 ( asK > 4 )$

Hence by $\text{PMI,}\text{P} \left(\text{n}\right)$ is true $\forall n \in \mathbb{N}, n ≥ 4$