Question

The exponent of $18$ in $200!$, is

• A. $24$
• B. $46$
• C. $47$
• D. $48$

Answer 1

The correct option is D $48$

Exponent of $18$ in $200!$

$\Rightarrow 18=(3{)}^2\times 2$

Exponent of $P$ in $n!$

$=\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+.............\left[\frac{n}{p^K}\right]=P^K\le n$

Exponential of $3$ in $n!$

$=\left[\frac{200}{3}\right]+\left[\frac{200}{9}\right]+\left[\frac{200}{27}\right]+\left[\frac{200}{81}\right]=66+22+4+2=94$

Exponent of $(3{)}^2=48$

Exponent of $2$ in $n!$

$=\left[\frac{200}{2}\right]+\left[\frac{200}{4}\right]+\left[\frac{200}{8}\right]+\left[\frac{200}{16}\right]>>48$

Least exponent $=48$

Exponential of $18$ is $48$

Hence the correct answer is $48$