Find n if ^ 2 n C_3: ^n C_2=52: 3

Question

Find n if ${ }^{2 n} C_3:{ }^n C_2=52: 3$

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$\begin{array}{ll} & { }^{2 n} C_3:{ }^n C_2=52: 3 \\ \therefore \quad & \frac{(2 n) !}{3 !(2 n-3) !} \div \frac{n !}{2 !(n-2) !}=\frac{52}{3} \\ \therefore \quad & \frac{(2 n) !}{3 !(2 n-3) !} \times \frac{2 !(n-2) !}{n !}=\frac{52}{3} \\ \therefore \quad & \frac{(2 n)(2 n-1)(2 n-2)(2 n-3) !}{3 \times 2 !(2 n-3) !} \times \frac{2 !(n-2) !}{n(n-1)(n-2) !}=\frac{52}{3} \\ \therefore \quad & \frac{2 n(2 n-1) \cdot 2(n-1)}{3} \times \frac{1}{n(n-1)}=\frac{52}{3} \\ \therefore \quad & \frac{4(2 n-1)}{3}=\frac{52}{3} \\ \therefore \quad & 2 n-1=13\end{array}$

$\begin{array}{ll}\therefore & 2 n=14 \\ \therefore & n=7\end{array}$

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