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PS - combination

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abhasjha GMAT Destroyer! Default Avatar
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PS - combination Post Sun Nov 16, 2014 1:17 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    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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    Post 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.

    Answer choice A (4 letters)
    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

    Answer choice B (5 letters)
    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

    Answer: B

    Cheers,
    Brent

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    Post 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.

    Answer choice A (4 letters)
    Let the letters be A, B, C, D
    The possible codes are:
    A
    B
    C
    D
    AB
    AC
    AD
    BC
    BD
    CD
    TOTAL = 10 (not enough. We need at least 12 codes)

    Answer choice B (5 letters)
    Let the letters be A, B, C, D, E
    The possible codes are:
    A
    B
    C
    D
    E
    AB
    AC
    AD
    AE
    BC
    BD
    BE
    CD
    CE
    DC
    TOTAL = 15

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

    Answer: B

    Cheers,
    Brent

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    Mathsbuddy GMAT Destroyer! Default Avatar
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    Post 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.
    Answer = B) 5.

    Mathsbuddy GMAT Destroyer! Default Avatar
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    Post 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.
    Answer = B) 5.
    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
    Answer = B) 5

    GMATinsight GMAT Destroyer!
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    Post 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

    Answer: Option B

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    Gurpreet singh Rising GMAT Star Default Avatar
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    Post Sun Jun 26, 2016 11:17 pm
    Brute force

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

    Alphabets used 5

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