combinatorics, you can win prizes from different categories but not from the same one. The question goes like this:

There is a conference for psychology in which $12$ researchers are participating. In the conference, two different companies are giving out prizes in two different categories. Three researchers will receive an award for "the most innovative research" (TV, DVD, or a radio), while two researchers will receive an award "the best presentation" (cash prize overall 10,000 dollars). If you know that every researcher can win in more than one category but can't win more than one prize in each category, how many different ways are there to divide the prizes?

choose one:
A) $14520$
B) $87120$
C) $95040$
D) $174240$
What I did was say each one of the researchers can win one prize of the first category, that's $C(12,3)$, then each one of the researchers can also win the second category, which is $C(12,2)$, which got me to $14520$, but apparently the answer is $87120$.

The first three prizes are distinct (TV, DVD, radio), so we use permutation instead of combinaison. There are $P(12,3)=1320$ ways to give the first three prizes.
The second prizes were the same, si combinaison was right here. There are $C(12,2)=66$ ways the give the second prizes.
Finally, there are $P(12,3)\times C(12,2)=87120$ wyas to give all the prizes.
