Tìm giá trị $\Large n \in \mathbb N$ thỏa mãn $\Large C_{n}^{6}+3 C_{n

Tìm giá trị $\Large n \in \mathbb N$ thỏa mãn $\Large C_{n}^{6}+3 C_{n}^{7}+3 C_{n}^{8}+C_{n}^{9}=2 C_{n+2}^{8}$

• A.

  $\Large n=18$

• B.

  $\Large n=16$

• C.

  $\Large n=15$

• D.

  $\Large n=14$

Điều kiện: $\Large n \geq 9$ và $\Large n \in \mathbb N$

Áo dụng công thức $\Large C_{n}^{k}+C_{n}^{k+1}=C_{n+1}^{k+1}$, ta có

$\Large C_{n}^{6}+3 C_{n}^{7}+3 C_{n}^{8}+C_{n}^{9}=2 C_{n+2}^{8}$

$\Large \Leftrightarrow C_{n}^{6}+C_{n}^{7}+2\left(C_{n}^{7}+C_{n}^{8}\right)+C_{n}^{8}+C_{n}^{9}=2 C_{n+2}^{8}$

$\Large \Leftrightarrow C_{n+1}^{7}+2 C_{n+1}^{8}+C_{n+1}^{9}=2 C_{n+2}^{8}$

$\Large \Leftrightarrow\left(C_{n+1}^{7}+C_{n+1}^{8}\right)+\left(C_{n+1}^{8}+C_{n+1}^{9}\right)=2 C_{n+2}^{8}$

$\Large \Leftrightarrow C_{n+2}^{8}+C_{n+2}^{9}=2 C_{n+2}^{8}$

$\Large \Leftrightarrow C_{n+2}^{9}=C_{n+2}^{8} \rightarrow n+2=9+8 \Leftrightarrow n=15$

Chọn C