Question

The product of the first $100$ odd natural numbers is

• A. $\frac{199!}{2^{100}\cdot 100!}$
• B. $\frac{200!}{2^{100}\cdot 100!}$
• C. $\frac{201!}{2^{100}\cdot 100!}$
• D. $\frac{100!}{2^{100}\cdot 100!}$

Answer 1

The correct option is B $\frac{200!}{2^{100}\cdot 100!}$

We know that, $\left(2n\right)!=2^n\cdot n!(1\times 3\times 5\times 7\times \cdots \times 2n-1)$
Here, $n=100$
$\Rightarrow 1\times 3\times 5\times 7\times \cdots \times 2n-1=\frac{\left(2n\right)!}{2^n\cdot n!}=\frac{200!}{2^{100}\cdot 100!}$