Find n from the equation

Question

How do you solve the below equation? The question read " There are n different dolls in a bag. The number of ways of choosing 4 dolls is same as 8 dolls. Find n. I came down till this below step and now i'm stuck.
(n-4)!=1680(n-8)!

Answer 1

easiest solution I know is to note $\binom{n}{k}=\binom{n}{n-k}$ this translates as $\binom{n}{4}=\binom{n}{n-4}$, we were told $\binom{n}{4}=\binom{n}{8}$ therefore $\binom{n}{8}=\binom{n}{n-4}$ which leads to 8=n-4 which then leads to n=12.

Answer 2

Here is where you will have to use your common sense.

$(n-4)(n-5)(n-6)(n-7) = 8.7.6.5$

How you get the above,

$\frac{(n-4)!}{(n-8)!} = (n-4)(n-5)(n-6)(n-7) = 1680 = 8.7.6.5$

For n to be integer, $n-4 = 8$ and thus $n = 12$