$101$
$99$
$100$

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$\begin{aligned} &= \dfrac{\big(100! + 99!\big) \times \big(98! + 97!\big) \times \ldots \big(2! + 1!\big)}{\big(100! - 99!\big) \times \big(98! - 97!\big) \times \ldots \big(2! - 1!\big)} \\ \\ &= \dfrac{\big(100 \times 99! + 99!\big) \times \big(98 \times 97! + 97!\big) \times \ldots \big(2 \times 1! + 1!\big)}{\big(100 \times 99! - 99!\big) \times \big(98 \times 97! - 97!\big) \times \ldots \big(2 \times 1! - 1!\big)} \\ \\ &= \dfrac{\cancel{99!} (100+1)}{\cancel{99!} (100-1)} \cdot \dfrac{\cancel{97!} (98+1)}{\cancel{97!} (98-1)} \cdots \dfrac{\cancel{1!} (2+1)}{\cancel{97!} (2-1)} \\ \\ &= \dfrac{101}{\cancel{99}} \cdot \dfrac{\cancel{99}}{\cancel{97}} \cdots \dfrac{\cancel{3}}{1} \\ \\ &= \boxed{101} \end{aligned}$