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Combinations , permutations ?

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sapuna Rising GMAT Star
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Combinations , permutations ? Post Mon Aug 11, 2014 6:54 am
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  • Lap #[LAPCOUNT] ([LAPTIME])
    A ceirtain stock exchange designates each item with a one , two or three letter code where each letter is selected from the 26 letters of the alphabet. If the letters may be repeated and if the same letter used in a different order constitue a different code, how many different stocks is it possible to uniquely designate with these codes ?

    2951

    8125

    15600

    16302

    18278

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    GMATinsight GMAT Destroyer!
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    Post Mon Aug 11, 2014 7:04 am
    sapuna wrote:
    A ceirtain stock exchange designates each item with a one , two or three letter code where each letter is selected from the 26 letters of the alphabet. If the letters may be repeated and if the same letter used in a different order constitue a different code, how many different stocks is it possible to uniquely designate with these codes ?

    2951

    8125

    15600

    16302

    18278
    Single Letter Code = 26
    Double Letter Code = 26 x 26 = 676
    Three Letter Code = 26 x 26 x 26 = 17576

    Total Such Codes = 17576 + 676 + 26 = 18278

    Answer: Option E

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    sapuna Rising GMAT Star
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    Post Mon Aug 11, 2014 7:32 am
    I thought as much. First, I thought that the correct score should end in what 6 multiplied by 6 is..which is 6 but there is no such answer. Then I figured I we have to add them and not multiply.

    Post Mon Aug 11, 2014 9:44 am
    Quote:
    A certain stock exchange designates each stock with a one-, two- or three-letter code, where each letter is selected from the 26 letters of the alphabets. If the letter maybe repeated and if the same letters used in different order constitude a different code, how many different stock is it possible to uniquely designate with these codes?

    A. 2,951
    B. 8,125
    C. 15,600
    D. 16,302
    E. 18,278
    My approach is similar to that of Bhoopendra, with a TWIST at the end.

    1-letter codes
    26 letters, so there are 26 possible codes

    2-letter codes
    There are 26 options for the 1st letter, and 26 options for the 2nd letter.
    So, the number of 2-letter codes = (26)(26) = 26²

    3-letter codes
    There are 26 options for the 1st letter, 26 options for the 2nd letter, and 26 options for the 3rd letter.
    So, the number of 3-letter codes = (26)(26)(26) = 26³

    So, the TOTAL number of codes = 26 + 26² + 26³

    IMPORTANT: Before we perform ANY calculations, we should first look at the answer choices, because we know that the GMAT test-makers are very reasonable, and they don't care whether we're able make long, tedious calculations. Instead, the test-makers will create the question (or answer choices) so that there's an alternative approach.

    The alternative approach here is to recognize that:
    26 has 6 as its units digit
    26² has 6 as its units digit
    26³ has 6 as its units digit

    So, (26)+(26²)+(26³) = (26)+(___6)+(____6) = _____8

    Since only E has 8 as its units digit, the answer must be E

    Cheers,
    Brent

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