Prove that \( { }^{2 n} C_{n}=\frac{2^{n}\{1 \cdot 3 \cdot 5 \ldots(2 n-1)\}}{n !} \).

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Prove that \( { }^{2 n} C_{n}=\frac{2^{n}\{1 \cdot 3 \cdot 5 \ldots(2 n-1)\}}{n !} \).
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