I want to simplify the expression $\frac{25!}{3!}$. I understand that it doesn't make sense to do that because it's already simplified but can the answer be found out without writing all of the numbers out?
The easiest way to "solve" would be to write the whole thing out by hand and use a calculator to solve this problem but in a non-calculator paper, that would be impossible. I'm looking for a more efficient way to solve this question.
I've tried the easy way:
It's $$\frac{25!}{3!}=\frac{25*24*23*22*21*20*...*6*5*4*3*2*1}{3*2*1}=\frac{1.55...*10^{25}}{6}=2.85...*10^{24}$$
I think to simplify it, it would be $\frac{25!}{3!}=25!±6$ because you don't want to multiply the last three digits, $3,2,1$, of $25!$ But that doesn't give the right answer.
Can this be simplified any further?
$\dfrac{25!}{3 !}= \dfrac{25\times 24\times23!}{6}= 100\times 23!$
Do you want its approximate value? If yes, you may then use the Stirling's formula:
$$n!\approx n^ne^{-n}\sqrt{2 \pi n} $$