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A limit of sums of binomial coefficients

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Post Tue May 24, 2016 3:10 pm

Posts: 99
Define binomial coefficients for $a\le R$ by
\[\left( {\begin{array}{*{20}{c}}
{a - 1} \\
i \\
\end{array}} \right) = \frac{{\left( {a - 1} \right)\left( {a - 2} \right) \cdots \left( {a - i} \right)}}{{i!}}.\]
How to solve the following limit
\[\mathop {\lim }\limits_{a \to m} \sum\limits_{1 \le {l_1} < {l_2} < \cdots {l_p} \le i} {\frac{{\left( {\begin{array}{*{20}{c}}
{a - 1} \\
i \\
\end{array}} \right)}}{{\left( {a - {l_1}} \right)\left( {a - {l_2}} \right) \cdots \left( {a - {l_p}} \right)}}} = ?\;\left( {0 \le m \in Z} \right)\]

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