Forming 4 groups from 16 people How many ways can $16$ students split up into $4$ study groups of size $4$ if (i) each group studies a different topic? (ii) all groups study the same topic?
My Idea : (i) Since with respect to topics we can form groups we see for the first topic there are $\binom{16}{4}$ ways next there are $\binom{12}{4}$ ways and next there are $\binom{8}{4}$ ways and the last $4$ gets fixed. So in total there are $\binom{16}{4}\binom{12}{4}\binom{8}{4}$ ways to form the scenario.
(ii) This will have the same ans as (i), since essentially we are forming groups. 
Is my idea correct? Or we need to consider the $4!$ term in the first part for permuting topics after forming groups? 

You are very close. The difference between (i) and (ii) is that the order of the four groups no longer matters (since they are all studying the same topic), so you need to divide by another 4!.  Hence the answer to (ii) is $\dfrac{\binom{16}{4}\binom{12}{4}\binom{8}{4}}{4!}$.
