Prove that : \( { }^{\mathrm{n}} C_{0} \cdot{ }^{2 n} C_{n}-{ }^{n} C_{1} \cdot{ }^{2 n-2} C_{n}...

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Prove that : \( { }^{\mathrm{n}} C_{0} \cdot{ }^{2 n} C_{n}-{ }^{n} C_{1} \cdot{ }^{2 n-2} C_{n}+{ }^{\mathrm{n}} C_{2} \cdot{ }^{2 n-4} C_{n}+\ldots=2^{n} \)
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