$$\begin{align}(n+1)^{p+1} - n^{p+1} &= \left(n^{p+1} + \dbinom{p+1}{1}n^p + \cdots + 1\right) - n^{p+1}\\
&= \dbinom{p+1}{1}n^p + \cdots + 1 \\
&> \dbinom{p+1}{1}n^p & (\because n > 0) \\
&= (p+1)n^p.\end{align}$$
Thus,
$$n^p < \dfrac{(n+1)^{p+1} - n^{p+1}}{p+1}.$$
So option (b) is correct