조합으로 제곱합 증명

다음은 등식 ${}_{n} \mathrm{C} \mathit{\mathrm{_{r}}} + _{n} \mathrm{C} \mathit{\mathrm{_{r + 1}}} = _{n + 1} \mathrm{C} \mathit{\mathrm{_{r + 1}}}$을 이용하여 $\displaystyle 1 ^{2} + 2 ^{2} + 3 ^{2} + \cdots + n ^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ ($n = 1 , 2 , 3 , \cdots$) 을 증명한 것이다.

<증명> $2$ 이상인 자연수 $k$에 대하여 $k ^{2} = \square {( \text{가} )}$ $+ 2 \cdot _{k} \mathrm{C} \mathit{_{2}}$로 나타낼 수 있으므로 $1 ^{2} + 2 ^{2} + 3 ^{2} + \cdots + n ^{2}$ $= _{1} \mathrm{C} _{1} + ( _{2} \mathrm{C} _{1} + 2 \cdot _{2} \mathrm{C} _{2} ) + ( _{3} \mathrm{C} _{1} + 2 \cdot _{3} \mathrm{C} _{2} ) + \cdots + ( _{\mathit{n}} C _{1} + 2 \cdot \square {( \text{나} )} )$ $= ( _{1} \mathrm{C} _{1} + _{2} \mathrm{C} _{1} + _{3} \mathrm{C} _{1} + \cdots +$${}_{n} \mathrm{C} \mathit{_{1}} )$$+ 2 ( _{2} \mathrm{C} _{2} + _{3} \mathrm{C} _{2} + \cdots + \square {( \text{나} )}$ $)$ $= _{n + 1} \mathrm{C} \mathit{_{2}} + 2 \cdot \square {( \text{다} )}$ $\displaystyle = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

위 증명에서 (가), (나), (다)에 알맞은 것은? [3점] (가) (나) (다) ①${}_{k} \mathrm{C} \mathit{_{1}}$ ${}_{n} \mathrm{C} \mathit{_{2}}$ ${}_{n} \mathrm{C} \mathit{_{3}}$②${}_{k} \mathrm{C} \mathit{_{1}}$ ${}_{n} \mathrm{C} \mathit{_{2}}$ ${}_{n + 1} \mathrm{C} \mathit{_{3}}$③${}_{k} \mathrm{C} \mathit{_{1}}$ ${}_{n + 1} \mathrm{C} \mathit{_{2}}$ ${}_{n} \mathrm{C} \mathit{_{3}}$④${}_{k + 1} \mathrm{C} \mathit{\mathrm{_{1}}}$ ${}_{n} \mathrm{C} \mathit{_{2}}$ ${}_{n} \mathrm{C} \mathit{_{3}}$⑤${}_{k + 1} \mathrm{C} \mathit{\mathrm{_{1}}}$ ${}_{n + 1} \mathrm{C} \mathit{_{2}}$ ${}_{n + 1} \mathrm{C} \mathit{_{3}}$