Twelve contestants at the county fair have entered their cakes

The answer key states that the order matters and thus it is correct to use a permutation instead of a combination. Can someone explain the reason the order matters here - I'm a little confused. Thanks.

Good question. Look below for the explanation.

Order matters here as you have to give out 4 UNIQUE prizes to 4 contestants.

Let us assume that the 4 winners are A,B,C and D. If this was a question in which the 4 prizes were given irrespective of the position, then the order would not have mattered. But as the color of the ribbon awarded is dependent on the position, the order matters for this question.

Start with awarding in this ORDER:

1st - A (purple)
2nd - B (blue)
3rd - C (red)
4th - D (white). This is your 1 of the possible combinations.

Now if A and D swap their places, then A no longer gets the purple ribbon while D does not get white. This new combination is

1st - D (purple)
2nd - B (blue)
3rd - C (red)
4th - A (white),

Another possible combination is

1st - A (purple)
2nd - C (blue)
3rd - B (red)
4th - D (white) etc

So you see that the order of selection matters as this order will determine what color is assigned to the winners.

Hope this helps.

P.S. do note that in the case that all 4 winners were awarded only 1 color ---> then the order of selection would not have made any difference.

The REASON is that all prizes are distinct..
a contestant say A getting a purple ribbon is different from his getting red ribbon..
so if 4 person are given these four different ribbons. each can get any of the four colured ribbon and will form a different arrangement..

Ways to select four winners from 12 contestants is 12C4 = 12!/8!*4!

Now anyone of the selected four could be at first position and similarly in 2nd, 3rd and 4th positions. So ways of distributing ribbons to winners is 4!

Final answer - (12!/8!*4!)*4! = 12!/8!.

Therefore D.