scary product !!!

$\frac { 2\times 4\times 6\times ....\times 100 }{ 1\times 3\times 5\times .....\times 99 }$ Now, we multiply $2*4*6*.....*100$ in both numerator and denominator to get $\frac { { (2\times 4\times 6\times ....\times 100) }^{ 2 } }{ 1\times 2\times 3\times 4\times 5\times .....\times 99\times 100 } \\ =\frac { { 2 }^{ 100 }\times 50!\times 50! }{ 100! } \\ =\frac { { 2 }^{ 100 } }{ \left( \begin{matrix} 100 \\ 50 \end{matrix} \right) }$ Using the asymptotic properties of central binomial coefficient, we know $\left( \begin{matrix} 2n \\ n \end{matrix} \right) \simeq \frac { { 2 }^{ 2n } }{ \sqrt { n\pi } }$ So rearranging the above gives $\frac { { 2 }^{ 100 }\sqrt { 50\pi } }{ { 2 }^{ 100 } } \\ =\sqrt { 50\pi } \\ =\sqrt { 157.08 } \\ \approx 12.5$ Hence, greatest integer value is $12$