Twelve contestants at the county fair have entered their cakes

Question

Twelve contestants at the county fair have entered their cakes to be judged in the cake decorating competition. A purple ribbon, blue ribbon, red ribbon, and white ribbon will be given to the first, second, third, and fourth place competitors, respectively. How many different ways are there to award the four ribbons to the contestants?

a)8!/(4!*4!)

b)12!/(8!*4!)

c)8!/4!

d)12!/8!

e)12!/4!

a)8!/(4!*4!)

b)12!/(8!*4!)

c)8!/4!

d)12!/8!

e)12!/4!

EMPOWERgmatRichC · Accepted Answer

Hi anurag16,

This question actually involves some fairly straight-forward math, but the answer choices require that you do a little extra 'work' to convert the correct answer into one of those formats.

Although the prompt doesn't state it, we're meant to assume that each of the 12 contestants enter JUST 1 CAKE each. A prize is given for 1st, 2nd, 3rd and 4th place. We're asked for the number of different arrangements of those 4 prizes.

For 1st place, there are 12 options....
Once that award is given, there are 11 options for 2nd place...
Once that award is given, there are 10 options for 3rd place...
Once that award is given, there are 9 options for 4th place...

So there are (12)(11)(10)(9) total options. However, the answer choices are NOT written in this format, so we have to look for an 'equivalent' to this number. Since the answer are all written with factorials, we have to find one with a factorial in the denominator that would 'cancel out' the proper part of the numerator. That happens with 12!/8!.

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

ENGRTOMBA2018 · Answer

The mistake you are doing is that you are neglecting the 4! ways in you can arrange 4 contestants for the 4 prizes.

Number of ways you can select 4 people out of 12 = 12C4

Once you select the 4 people, you have the following arrangement, PBRW (PBRW being the 4 prizes) but the same group of people can also be chosen against BRWP etc. Thus you get 4! ways of arranging 4 prizes.

Thus total possible ways = 12C4*4! = 12!/8!. D is the correct answer.

Hope this helps.

jn30 · Answer

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.

ENGRTOMBA2018 · Answer

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.

chetan2u · Answer

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..

poorvi125 · Answer

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.

luisdicampo · Answer

Deconstructing the Question  
There are  12 distinct contestants  and  4 distinct ribbons  corresponding to first, second, third, and fourth place.
Because each ribbon represents a different rank,  order matters .
This is a permutation problem, not a combination.

Step-by-step  
The first-place ribbon can be awarded to any of the  12  contestants.
The second-place ribbon can be awarded to any of the remaining  11  contestants.
The third-place ribbon can be awarded to any of the remaining  10  contestants.
The fourth-place ribbon can be awarded to any of the remaining  9  contestants.

Multiplying these choices gives  12 	imes 11 	imes 10 	imes 9 = \frac{12!}{8!} .

Answer:  12!/8!

Twelve contestants at the county fair have entered their cakes

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