Please explain what the difference between these two problems is. Why does one use the n!/(n-k)! and the other uses n!/[k!(n-k)!. The problems seem the same to me. Thank you!
in the first problem, order matters.
12 possibilities for the winner
11 possibilities for 2nd place
10 possibilities for 3rd place
if the winning employees were a, b, and c in that order, that would not be the same as b, c, and a in that order.
when order matters, you are dealing with a permutation.
in the second problem, order does not matter ...
nurses a, b, and c are in the same group as nurses b, c, and a
when order does not matter, you are dealing with a combination.