Explanation:

Let us see this. We know that 4! =24 which is divisible by 4.
Now, after 4!,we can write each factorial term with as 4! * y where y is a positive number.
ex - 5! = 4! *5
 6! = 4! * 5 * 6.
So 4! is a factor of 5!,6!,7!, ... ,100!.
So , from 4! to 100!, each term is divisible by 4!(24) as they have 4! as their factor.
So, 4! mod 24 = 0.
Similarly , 5! %24 =0 and so for rest upto 100!.
Now, Rule of modular summation states that
(a+b)%c = (a%c + b%c) %c.
Now,

(1! +2! +3! +4! ...+100!) mod 24 =

(1!mod24 + 2!mod24 + 3!mod 24 + 4! mod24 + 5!mod24 + 6!mod24 +... + 100!mod24)%24
=( 1 mod 24 + 2 mod 24 + 6 mod 24 + 0 + 0 +0 .. +0)mod 24
= (1 + 2 +6) mod 24
= 9 mod 24
=9.

Hence, option 3 is the correct answer.