I was given this problem:
"How many ways can you arrange 16 people into 4 rows of 4 desks each?"
and the answer is $16!$ simply based on the idea that every desk is uniquely identifiable and we have 16 desks.
But, initially, I had approached the problem another way. First, the number of possible groups of 4 people we can get is $\binom{16}{4}$. Then the number of assignments we can do in order to match groups to rows is 4! and finally the rearrangements within a row are also 4! yielding in total: $\binom{16}{4}4!4!=16\cdot 15\cdot 14\cdot 13\cdot 4!$ which is different. What am I doing wrong?
The binomial coefficient $\binom{16}{4}$ is the number of ways to select only one group of $4$. Following your approach, the correct counting should be $$\binom{16}{4}\cdot \binom{12}{4}\cdot \binom{8}{4}\cdot \binom{4}{4}\cdot (4!)^4.$$