STD 11 - 6. permutation and combination — JEE STD 11 Science — Question

Exponent of $2$ in $200 !$.

$=\left[\begin{array}{c}200 \\ 2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^3\end{array}\right]+\left[\begin{array}{c}200 \\ 2^4\end{array}\right]+\left[\begin{array}{c}200 \\ 2^5\end{array}\right] +\left[\frac{200}{2^8}\right]+\left[\begin{array}{c}200 \\ 2^7\end{array}\right]+\left[\begin{array}{c}200 \\ 2^8\end{array}\right]$

$=100+50+25+12+6+3+1=197$

Exponent of $2$ in $100!$

$=\left[\frac{100}{2}\right]+\left[\frac{100}{2^2}\right]+\left[\begin{array}{c}100 \\ 2^3\end{array}\right]+\left[\begin{array}{c}100 \\ 2^4\end{array}\right]+\left[\begin{array}{c}100 \\ 2^5\end{array}\right]+\left[\begin{array}{c}100 \\ 2^6\end{array}\right]+\left[\begin{array}{c}100 \\ 2^7\end{array}\right]$

$\operatorname{In} \frac{200 !}{100 !}=\frac{2^{197}}{2^{97}}=2^{100}$

$\therefore$ The largest power of $2$ is $100$.