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The $i^{th}$ elementary symmetric polynomial of the set of $n$ variables$\mathcal{R}=\{m_1,m_2,m_3,...,m_n\}$ is isolated from the expansion of the$i^{th}$ binomial product ${{m_1+...+m_n}\choose i}$ via an alternating sum.

Extraction of the $i^{th}$ Elementary Symmetric Polynomial from a Product in Binomial Form ${{m_1+...+m_n}\choose i}$