(C) We are given the statement $P(n): 2^{n} < n!$. We test for positive integer values of $n$: For $n=1$: $2^{1} = 2$ and $1! = 1$. Since $2 < 1$ is f

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(C) We are given the statement $P(n): 2^{n} We test for positive integer values of $n$: For $n=1$: $2^{1} = 2$ and $1! = 1$. Since $2 For $n=2$: $2^{2} = 4$ and $2! = 2$. Since $4 For $n=3$: $2^{3} = 8$ and $3! = 6$. Since $8 For $n=4$: $2^{4} = 16$ and $4! = 24$. Since $16 Therefore,the smallest positive integer for which $P(n)$ is true is $n=4$.