The following proves by mathematical induction that for all natural numbers $n$,

$$\left( 1 ^ { 2 } + 1 \right) \cdot 1 ! + \left( 2 ^ { 2 } + 1 \right) \cdot 2 ! + \cdots + \left( n ^ { 2 } + 1 \right) \cdot n ! = n \cdot ( n + 1 ) !$$

holds.

\section*{<Proof>}
(1) When $n = 1$, (left side) = 2, (right side) = 2, so the given equation holds.\\
(2) Assuming it holds when $n = k$,

$$\begin{aligned}
\left( 1 ^ { 2 } + 1 \right) \cdot 1 ! & + \left( 2 ^ { 2 } + 1 \right) \cdot 2 ! + \cdots \\
& + \left( k ^ { 2 } + 1 \right) \cdot k ! = k \cdot ( k + 1 ) !
\end{aligned}$$

We show that it holds when $n = k + 1$.

$$\begin{aligned}
\left( 1 ^ { 2 } + 1 \right) \cdot 1 ! + \left( 2 ^ { 2 } + 1 \right) \cdot 2 ! + \cdots + \left( k ^ { 2 } + 1 \right) \cdot k ! + \left\{ ( k + 1 ) ^ { 2 } + 1 \right\} \cdot ( k + 1 ) ! \\
= ( \text{ (A) } ) + \left\{ ( k + 1 ) ^ { 2 } + 1 \right\} \cdot ( k + 1 ) ! \\
= ( k + 1 ) \cdot \text{((B))} \cdot ( k + 1 ) ! \\
= \text{ ((C)) }
\end{aligned}$$

Therefore, it also holds when $n = k + 1$.\\
Thus, the given equation holds for all natural numbers $n$.

Which expressions are correct for (A), (B), and (C) in the above proof? [3 points]

\begin{center}
\begin{tabular}{ l c c c }
 & $\underline { ( \text{ (A) } ) }$ & $\underline { ( \text{ (B) } ) }$ & $\underline { ( \text{ (C) } ) }$ \\
$( 1 ) k \cdot ( k + 1 ) !$ & $k ^ { 2 } + 2 k + 1$ & $( k + 1 ) !$ &  \\
$( 2 ) k \cdot ( k + 1 ) !$ & $k ^ { 2 } + 3 k + 2$ & $( k + 2 ) !$ &  \\
$( 3 ) k \cdot ( k + 1 ) !$ & $k ^ { 2 } + 3 k + 2$ & $( k + 1 ) !$ &  \\
$( 4 ) ( k + 1 ) \cdot ( k + 1 ) !$ & $k ^ { 2 } + 3 k + 2$ & $( k + 2 ) !$ &  \\
$( 5 ) ( k + 1 ) \cdot ( k + 1 ) !$ & $k ^ { 2 } + 2 k + 1$ & $( k + 1 ) !$ &  \\
\end{tabular}
\end{center}