color coding

[ritula]

John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? Assume that changing the color order within a pair does not produce different codes.

24
12
7
6
5

[mals24]

IMO E

The question is asking what is the minimum number of combination of 1 paint or 2 paints possible

i.e. its asking for the minimum value for xC1+xC2>12-----equ1

Now the combination has to be greater than 12 since that is the total number of clients

The answer is 5

5C1 + 5C2 = 5+10 = 15

You can clearly rule out 12 and 24 since they will give you really high values for equ1.

[stop@800]

yes, it should be 5 colors

[ssy]

Mals24 - I'm not sure if I understand your explanation. Could you please clarify? What do these equations mean? Thanks!

xC1+xC2>12
5C1 + 5C2 = 5+10 = 15

[mals24]

@ssy

The question is asking us what is the "minimum" number of colors needed for coding. And we can either assign a single color to a client or use two different colors.

In other words the question is asking us what is the minimum number of combination of one or two color codes.

So we set up the equation nC1+nC2>12 (this requires the use of combination and not permutation since the order of colors does not matter)

nC1 is the number of ways we can assign one color to the client
nC2 is the number of ways we can assign two diff colors to the client
+ since its "or"
">12" since the combination should be greater than 12
It cannot be less than 12 since we have a total of 12 clients and we have to find the "minimum" number of color combination. Hence the greater sign

Now we need to see among the answer choices which choice when substituted in place of "n" gives us the least value in the equation.

I hope you get the logic

[Scott@TargetTestPrep]

John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? Assume that changing the color order within a pair does not produce different codes.

24
12
7
6
5

Solution:

Since there are only 12 clients, the number of colors needed is even fewer. Let’s check the answer choices starting with 5.

If there are 5 colors, the number of single-color codes is 5, and the number of codes that are pairs of two different colors is 5C2 = (5 x 4)/2 = 10. Therefore, if there are 5 colors, the total number of codes that can be formed is 5 + 15 = 15, which is more than sufficient since there are only 12 clients.

Answer: E