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Math Help - Selections. Why am I short?

  1. #1
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    Selections. Why am I short?

    Hi everyone,

    This is a spin off from another question.

    4 letters are chosen and repetition is allowed so there are $26^4= 456976$ selections possible.

    Now I am going to split this up.

    Letters all different = 26252423=358800
    letters 2 the same= 26125244C2=93600
    letters 3 the same=261125∗4C3=2600
    Letters all the same=26

    Add all these and I get 455026
    WHY AM I SHORT???

    Can anyone please tell me what selections I have missed out when I pulled apart the total?
    Last edited by Melody2; July 7th 2014 at 02:26 AM.
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  2. #2
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    Re: Selections. Why am I short?

    You're missing combos with 2 pairs of letters

    There are 26*25 ways to pick the letters and

    $\dfrac 1 2 \begin{pmatrix}4 \\2\end{pmatrix}$ ways of arranging them in 4 slots.

    The $\dfrac 1 2$ is because within the pair the letters can be flipped and you obtain the same code.

    Selections.  Why am I short?-clipboard01.jpg
    Thanks from emakarov, topsquark and Melody2
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  3. #3
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    Re: Selections. Why am I short?

    4 letters are chosen and repetition is allowed so there are 264=456976 selections possible.

    Now I am going to split this up.

    Letters all different = 26∗25∗24∗23=358800
    letters 2 the same (and two different)= 26∗1∗25∗24∗4C2=93600
    letters 2 pairs the same = 26*1*25*1*4C2/2 = 26*1*25*6/2=1950
    letters 3 the same=26∗1∗1∗25∗4C3=2600
    Letters all the same=26

    358800+93600+1950+2600+26=456976.

    That is excellent. Thank you Romsek!
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