Given : $\frac{1!+2!+3!+........+100!}{24}$

Now it can also be written as $\frac{1}{24}$ + $\frac{1*2}{24}$ + $\frac{1*2*3}{24}$ + $\frac{1*2*3*4}{24}$ + ............ + $\frac{1*2*3*4*5*6*......*100}{24}$

But 4! =24

Therefore every no. after 4! will give a quotient and will leave a remainder 0 ( For example 5! will give quotient 5 and remainder zero ).

This means that remainder will be equal to 1! + 2! + 3! which is $\boxed{1 + 2+ 6 = 9}$