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## PS - combination

tagged by: Brent@GMATPrepNow

This topic has 2 expert replies and 4 member replies
abhasjha Master | Next Rank: 500 Posts
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#### PS - combination

Sun Nov 16, 2014 1:17 pm
A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4
B. 5
C. 6
D. 7
E. 8

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Brent@GMATPrepNow GMAT Instructor
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Sun Nov 16, 2014 2:13 pm
abhasjha wrote:
A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4
B. 5
C. 6
D. 7
E. 8
One approach is to add a BLANK to the letters in order to account for the possibility of using just one letter for a code.

ASIDE: Notice that, if we select 2 characters, there's only 1 possible code that can be created. The reason for this is that the 2 characters must be in ALPHABETICAL order. Or, in the case that a letter and a blank are selected, there's only one possible code as well.

Now we'll test the answer choices.

Let the letters be A, B, C, D
We'll add a "-" to represent a BLANK.
So, we must choose 2 characters from {A, B, C, D, -}
In how many ways can we select 2 characters?
We can use combinations to answer this. There are 5 characters, and we must select 2. This can be accomplished in 5C2 ways (= 10 ways).
So, there are only 10 possible codes if we use 4 letters. We want at least 12 codes.

ASIDE: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: http://www.gmatprepnow.com/module/gmat-counting?id=789

Let the letters be A, B, C, D, E
Once again, we'll add a "-" to represent a BLANK.
So, we must choose 2 characters from {A, B, C, D, E, -}
There are 6 characters, and we must select 2. This can be accomplished in 6C2 ways (= 15 ways...PERFECT).

So, the least number of characters needed is 5

Cheers,
Brent

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Brent@GMATPrepNow GMAT Instructor
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Sun Nov 16, 2014 2:17 pm
abhasjha wrote:
A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4
B. 5
C. 6
D. 7
E. 8
We can also TEST each answer choice by LISTING all possible codes.

Let the letters be A, B, C, D
The possible codes are:
A
B
C
D
AB
AC
BC
BD
CD
TOTAL = 10 (not enough. We need at least 12 codes)

Let the letters be A, B, C, D, E
The possible codes are:
A
B
C
D
E
AB
AC
AE
BC
BD
BE
CD
CE
DC
TOTAL = 15

Perfect, 5 letters will give us the 12 codes we need.

Cheers,
Brent

_________________
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Mathsbuddy Master | Next Rank: 500 Posts
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Wed Nov 19, 2014 5:44 am
Imagine a 2 digit number AB where A and B are integers >= 0
Let's see what is the lowest base that can generate the value twelve using just these 2 digits?
Using base 4, the maximum number achievable using 2 digits is 33 (base 4) = 3x4 + 3 = 15 (denary)
However we need to remove 3 doubles (where A=B): 11,22,33 which leaves us with 15-3 = 12
and also remove half of these (as AB is accepted, but not BA), leaving us with 6
We can also add 4 for the 4 single digits (0,1,2,3)
Therefore Base 4 produces 10 possibilities
Hence any greater base will suffice (as the next base will produce plenty more)
Therefore base 5 is the lowest base.

Mathsbuddy Master | Next Rank: 500 Posts
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Wed Nov 19, 2014 6:57 am
Mathsbuddy wrote:
Imagine a 2 digit number AB where A and B are integers >= 0
Let's see what is the lowest base that can generate the value twelve using just these 2 digits?
Using base 4, the maximum number achievable using 2 digits is 33 (base 4) = 3x4 + 3 = 15 (denary)
However we need to remove 3 doubles (where A=B): 11,22,33 which leaves us with 15-3 = 12
and also remove half of these (as AB is accepted, but not BA), leaving us with 6
We can also add 4 for the 4 single digits (0,1,2,3)
Therefore Base 4 produces 10 possibilities
Hence any greater base will suffice (as the next base will produce plenty more)
Therefore base 5 is the lowest base.
Using the same algorithm for base 5, we get:
Maximum number = 44 (base 5) = 4 x 5 + 4 = 24 (denary)
Remove 4 doubles (11, 22, 33, 44) -> 24 - 4 = 20
Half this to remove alphabetical reversals -> 10
Add 5 for the 5 single digits -> 10 + 5 = 15
15 > 12, so base 5 works

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GMATinsight Legendary Member
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Thu Nov 27, 2014 10:23 am
abhasjha wrote:
A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?

A. 4
B. 5
C. 6
D. 7
E. 8
A typical P&C approach

If n are the total letters then
total combination of 1 Letter code = n
total combination of 2 Letter code (in alphabetical order) = nC2 = n(n-1)/2

Therefore total combination = n + n(n-1)/2 which must be greater than or equal to 12
i.e. 2n+n^2-n >or= 24
i.e. n^2 +n >or= 24

i.e. Minimum value of n must be 5

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Gurpreet singh Senior | Next Rank: 100 Posts
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Sun Jun 26, 2016 11:17 pm
Brute force

1 A
2 B
3 AB
4 C
5 CA
6 CB
7 D
9 DB
10 DC
11 E
12 EA

Alphabets used 5

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