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01 Mar 2025, 03:20
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gmatophobia
Happy Monday Everyone !!
Let’s kill the Monday Blue with a P&C question (more questions on the way
)
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.
A. 24
B. 12
C. 7
D. 6
E. 5
Source: GMATClubTest | Difficulty Level : Medium
It's a combination problem because order doesn't matter.Let’s kill the Monday Blue with a P&C question (more questions on the way
)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.
A. 24
B. 12
C. 7
D. 6
E. 5
Source: GMATClubTest | Difficulty Level : Medium
XC2 >= 12
For X = 5 -> 5C2 = 10
For X = 6 -> 6C2 = 15
If you go with 5C2 + 2C1 = 12, you will need 7 colors
But if you go with 6C2 = 15, you will need just 6 colors though you will have extra pairs that will be unused.
So the answer is D. 6
01 Mar 2025, 03:20
Kudos
Bookmarks
gmatophobia
Happy Monday Everyone !!
Let’s kill the Monday Blue with a P&C question (more questions on the way )
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.
A. 24
B. 12
C. 7
D. 6
E. 5
Source: GMATClubTest | Difficulty Level : Medium
It's a combination problem because order doesn't matter.Let’s kill the Monday Blue with a P&C question (more questions on the way
)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.
A. 24
B. 12
C. 7
D. 6
E. 5
Source: GMATClubTest | Difficulty Level : Medium
XC2 >= 12
For X = 5 -> 5C2 = 10
For X = 6 -> 6C2 = 15
If you go with 5C2 + 2C1 = 12, you will need 7 colors
But if you go with 6C2 = 15, you will need just 6 colors though you will have extra pairs that will be unused.
So the answer is D. 6
01 Mar 2025, 03:48
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Bookmarks
That question is discussed in detail here: john-has-12-clients-and-he-wants-to-use-color-coding-to-iden-107307.html
Official Solution:
John has 12 clients, and he wants to use color coding to identify each of them. He can use either a single color or a pair of two different colors to represent a client code. Assuming that switching the order of colors within a pair does not create a different code, what is the minimum number of colors needed for this coding scheme?
A. 24
B. 12
C. 7
D. 6
E. 5
Combination approach:
Let the number of colors needed be \(n\). Then, ensuring that the total number of unique color codes (single color codes + two-color codes) is sufficient for all 12 clients, the inequality \(n + C^2_n \ge 12\) must hold.
\(n+\frac{n(n-1)}{2} \ge 12\);
\(2n+n(n-1) \ge 24\);
\(n(n+1) \ge 24\). As \(n\) is an integer (it represents the number of colors), then \(n \ge 5\), so \(n_{\text{min}}=5\).
Trial and error approach:
If the minimum number of colors needed is 4, then there are 4 single color codes possible PLUS \(C^2_4=6\) two-color codes. Total: \(4+6=10 < 12\). Not enough for 12 codes;
If the minimum number of colors needed is 5, then there are 5 single color codes possible PLUS \(C^2_5=10\) two-color codes. Total: \(5+10=15 > 12\). More than enough for 12 codes.
As the least answer choice is 5, if you tried it first, you would arrive at the correct answer immediately.
Answer: E
Hope it’s clear.
Official Solution:
John has 12 clients, and he wants to use color coding to identify each of them. He can use either a single color or a pair of two different colors to represent a client code. Assuming that switching the order of colors within a pair does not create a different code, what is the minimum number of colors needed for this coding scheme?
A. 24
B. 12
C. 7
D. 6
E. 5
Combination approach:
Let the number of colors needed be \(n\). Then, ensuring that the total number of unique color codes (single color codes + two-color codes) is sufficient for all 12 clients, the inequality \(n + C^2_n \ge 12\) must hold.
\(n+\frac{n(n-1)}{2} \ge 12\);
\(2n+n(n-1) \ge 24\);
\(n(n+1) \ge 24\). As \(n\) is an integer (it represents the number of colors), then \(n \ge 5\), so \(n_{\text{min}}=5\).
Trial and error approach:
If the minimum number of colors needed is 4, then there are 4 single color codes possible PLUS \(C^2_4=6\) two-color codes. Total: \(4+6=10 < 12\). Not enough for 12 codes;
If the minimum number of colors needed is 5, then there are 5 single color codes possible PLUS \(C^2_5=10\) two-color codes. Total: \(5+10=15 > 12\). More than enough for 12 codes.
As the least answer choice is 5, if you tried it first, you would arrive at the correct answer immediately.
Answer: E
Hope it’s clear.
