GMAT Problem Solving - John has 12 Clients and He Wants to Use Color Coding to Identify Each Client

Question: 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.

1. 24
2. 12
3. 7
4. 6
5. 5

“John has 12 clients and he wants to use color coding to identify each client” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer: E
Solution and Explanation:
Approach Solution 1:
Combination approach -
In this question, it is given that John has 12 clients. He wants to assign code to each client. It is given that either a single color or a pair of two different colors can represent a client code. It has asked, what is the minimum number of colors needed for the coding?

Let us assume that the number of colors needed be n, then it must be true that n + 2Cn ≥ 12 (2Cn - number of ways to choose the pair of different colors from n colors when order doesn't matter).
n+n(n−1)/2 ≥12
2n + n(n−1) ≥ 24
n (n+1) ≥ 24. As n is an integer (it represents Number of colors), then n≥5, so nmin=5.
The correct answer is option E.

Approach Solution 2:
Trial and Error -  
In this question, it is given that John has 12 clients. He wants to assign code to each client. It is given that either a single color or a pair of two different colors can represent a client code. It has asked, what is the minimum number of colors needed for the coding?

If the minimum number of colors needed is 4 then there are 4 single color codes possible PLUS 2C4=6 two-color codes. Total: 4+6=10 < 12.
So this is not enough for 12 codes.
If the minimum number of colors needed is 5 then there are 5 single color codes possible PLUS 2C5 = 10 two-color codes. Total: 5+10=15 > 12.
This is more than enough for 12 codes.
Here the least choice is 5 then if you tried it first you'd get the correct answer.
Correct answer is E.

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