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Câu hỏi: Cho ${n \in \mathbb{N}*}$ và ${C_n^3 = A_n^2 - 10}$. Giá trị của ${n}$ là?
A. $\left[ \begin{matrix}
n=5 \\
n=6 \\
\end{matrix} \right. $.
B. $ \left[ \begin{matrix}
n=4 \\
n=6 \\
\end{matrix} \right. $.
C. $ \left[ \begin{matrix}
n=4 \\
n=5 \\
\end{matrix} \right. $.
D. $\left[ \begin{matrix}
n=4 \\
n=6 \\
\end{matrix} \right.$.
A. $\left[ \begin{matrix}
n=5 \\
n=6 \\
\end{matrix} \right. $.
B. $ \left[ \begin{matrix}
n=4 \\
n=6 \\
\end{matrix} \right. $.
C. $ \left[ \begin{matrix}
n=4 \\
n=5 \\
\end{matrix} \right. $.
D. $\left[ \begin{matrix}
n=4 \\
n=6 \\
\end{matrix} \right.$.
Ta có$\left\{ \begin{aligned}
& n\in \mathbb{N}* \\
& C_{n}^{3}=A_{n}^{2}-10 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \dfrac{n!}{3!\left( n-3 \right)}=\dfrac{n!}{\left( n-2 \right)!}-10 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \dfrac{n\left( n-1 \right)\left( n-2 \right)}{6}=n\left( n-1 \right)-10 \\
\end{aligned} \right.$
$\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& {{n}^{3}}-9{{n}^{2}}+8n+60=0 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \left[ \begin{aligned}
& n=-2 \\
& n=5 \\
& n=6 \\
\end{aligned} \right. \\
\end{aligned} \right. $ $ \Leftrightarrow \left[ \begin{aligned}
& n=5 \\
& n=6 \\
\end{aligned} \right.$
& n\in \mathbb{N}* \\
& C_{n}^{3}=A_{n}^{2}-10 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \dfrac{n!}{3!\left( n-3 \right)}=\dfrac{n!}{\left( n-2 \right)!}-10 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \dfrac{n\left( n-1 \right)\left( n-2 \right)}{6}=n\left( n-1 \right)-10 \\
\end{aligned} \right.$
$\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& {{n}^{3}}-9{{n}^{2}}+8n+60=0 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& n\ge 3,n\in \mathbb{N} \\
& \left[ \begin{aligned}
& n=-2 \\
& n=5 \\
& n=6 \\
\end{aligned} \right. \\
\end{aligned} \right. $ $ \Leftrightarrow \left[ \begin{aligned}
& n=5 \\
& n=6 \\
\end{aligned} \right.$
Đáp án A.