Prove that \( { }^{n} C_{1}\left({ }^{n} C_{2}\right)^{2}\left({ }^...

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Prove that \( { }^{n} C_{1}\left({ }^{n} C_{2}\right)^{2}\left({ }^{n} C_{3}\right)^{3} \) \( \cdot\left({ }^{n} C_{n}\right)^{n} \leq\left(\frac{2^{n}}{n+1}\right)^{n+1} C_{2} \)
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