<s>[INST] Approximate solution: factorial and exponentials If z= $\dbinom{200}{100}/(4^{100})$, what is the value of z?
The options are:
a. $z<1/3$
b. $1/3<z<1/2$
c. $1/2<z<2/3$
d. $2/3<z<1$
How should I go about solving these type of problems?
 [/INST] Here is an elementary approach without estimating the value of the expression:
\begin{eqnarray*} \frac{1}{4^m}\binom{2m}{m}
& = & \frac{1}{4^m}\cdot \frac{\prod_{i=1}^m 2i \cdot \prod_{i=1}^m (2i-1) }{(m!)^2} \\
& = & \frac{1}{4^m}\cdot 4^m\frac{\prod_{i=1}^m i \cdot \prod_{i=1}^m \left(i-\frac{1}{2}\right) }{\prod_{i=1}^m i \cdot \prod_{i=1}^m i} \\
& = & \prod_{i=1}^m \left( 1 - \frac{1}{2i} \right) \\
& = & \prod^m_{i=1} \frac{2i-1}{2i} \\
& \stackrel{m=100}{<} & \frac{1}{2}\cdot\frac{3}{4}\cdot \frac{5}{6} \\
& = & \frac{5}{16}\\
& < & \frac{1}{3}\\
\end{eqnarray*}
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