A number theory problem by Hemanth K

How many digits are there in the number $\dfrac{100!}{50!}$ when written in decimal representation?

You may find the following approximations useful.

• $5! = 120$ has 3 digits.
• $\log_{10} 2 \approx 0.3010$ .
• $\log_{10} 3 \approx 0.4771$ .
• $\log_{10} 7 \approx 0.8451$ .

Notation : $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .