A secretary differentiates files by a variety of color combinations.

Question

A secretary differentiates files by a variety of color combinations. If she uses 4 different colors, and marks each file with 1, 2, or 3 of the colors, how many different color combinations can she generate?

(A) 4
(B) 12
(C) 14
(D) 15
(E) 24

(A) 4 
(B) 12 
(C) 14 
(D) 15 
(E) 24

GMATinsight · Answer

If she marks by only one color then total possible combinations = 4C1 = 4

If she marks by only two colors then total possible combinations = 4C2 = 6

If she marks by only three colors then total possible combinations = 4C3 = 4

Total possible combinations = 4+6+4 = 14

Answer: Option C

BrentGMATPrepNow · Answer

What's the source of this question?
I ask because it doesn't specify whether the order of the colors matters.
For example, is the color combination RED-BLUE-YELLOW different from BLUE-YELLOW-RED?
Based on the official answer, it seems that the above combinations are the same, but how do we know this?

ASIDE: es, I know that some people will feel that the word "combination" in the question basically tells us that order does not matter. However, I'll argue that using the word "combination" does not necessarily imply that order does not matter....... unless we want to then conclude that it doesn't matter which order we enter the code numbers a combination lock

Cheers,
Brent

Cheers, Brent

energetics · Answer

Source is GMAT Hacks, going through it now (a lower quality, poor man's version of MGMAT, imo... although more concise in its explanations). Would an actual GMAT problem always state that colors are not ordered or something, rather just use the word combined or arranged?

BrentGMATPrepNow · Answer

The test-makers don't have one particular way to let test-takers know whether order matters. Sometimes the context of the question let's you know. Other times, there is clear wording (and sometimes even an example) that indicates whether order matters.

The important thing to keep in mind is that the test-makers take every precaution to minimize/eliminate any ambiguity.

Cheers,
Brent

DarkHorse2019 · Answer

We usually multiply the combinations when combining the result, why did we add here?

Please help  chetan2u ,  Gladiator59 ,  VeritasKarishma ,  Bunuel ,  generis

goeladit · Answer

Since this is an OR statement. She has three choices to mark the files i.e by one color or two or three colors.

KarishmaB · Answer

Partial solutions are multiplied (A AND B AND C) and complete solutions are added (X OR Y).

Let's understand this with an example:

Say there is a fast food joint which serves meals consisting of a burger/pizza slice, a side and a drink.
There are 3 different kids of burgers, 4 different kinds of pizza, 2 different sides and 5 different drinks available. 
In how many different ways can you make a meal?

Let's make burger meals first. We can choose a burger in 3 ways, sides in 2 ways and drinks in 5 ways. 
You need to choose a burger AND a side AND a drink.
So total burger meals = 3*2*5 = 30 burger meals

Now let's make pizza meals. We can choose a pizza slice in 4 ways, sides in 2 ways and drinks in 5 ways. 
You need to choose a pizza AND a side AND a drink.
So total pizza meals = 4*2*5 = 40 pizza meals

Now, to choose a meal, you pick either a burger meal OR a pizza meal. The 30 burger meals are complete in themselves and 40 pizza meals are complete in themselves too. 
So you ADD them to get 30 + 40 = 70 meals. You can pick a meal in 70 ways.

In the original question, she can mark a file using 1 colour only. So the 4 colours give us 4 different complete solutions. 
OR she can mark a file using 2 colours 4C2 = 6 ways of marking a file using 2 colours (complete solutions). 
Or she can mark a file using 3 colours in 4C3 = 4 ways. Again complete solutions

So we add them to get 4 + 6 + 4 = 14 solutions.

ScottTargetTestPrep · Answer

The total number of different color combinations that can be generated, assuming the order does not matter, is:

4C1 + 4C2 + 4C3 = 4 + 6 + 4 = 14

Answer: C

Rebaz · Answer

What about 4C4, she can also mark 4 colors, cann't she do that?

AnkithaSrinivas · Answer

she can use max of 3 colors

luisdicampo · Answer

Deconstructing the Question  
A file can be marked with  1, 2, or 3 colors  chosen from  4 colors .
Order does not matter → use combinations and add the cases.

Step-by-step

Choose 1 color:
 C(4,1)=4

Choose 2 colors:
 C(4,2)=6

Choose 3 colors:
 C(4,3)=4

Total combinations:
 4+6+4=14

Answer: 14

1111fate · Answer

if she chooses the same colour i.e 4 ways then how will she differentiate between the folders? they all will be same then! How to assume that ?

A secretary differentiates files by a variety of color combinations.

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A secretary differentiates files by a variety of color combinations.

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