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Math Help - Letter Combinations

  1. #1
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    Letter Combinations

    How many 3 letter combos can be made from the word proportion?
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  2. #2
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    Quote Originally Posted by thefirsthokage View Post
    How many 3 letter combos can be made from the word proportion?
    Number of letters is 10.

    And divide by repeats,
    \frac{10!}{2!2!3!1!1!1!}
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  3. #3
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    Hello, thefirsthokage!

    There is no neat formula for this one.
    I made a list . . . the shortest possible (I think).


    How many three-letter combos can be made from the word PROPORTION?
    We have these letters: . \begin{Bmatrix}O\,O\,O \\ P\,P \\ R\,R\\ T \\ I \\ N\end{Bmatrix}

    3 letters the same: OOO . . . one way.

    2 letters the same: there 3 choices of the matching pair ( OO,\,PP,\text{ or }RR)
    . . and 5 choices for the third letter . . . 3 \times 5 \:=\:15 ways.

    3 differerent letters: there are \binom{6}{3} = 20 ways.


    Therefore, there are: 1 + 15 + 20 \:=\:\boxed{36} three-letter combos.

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  4. #4
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    Here is a second way the get the same answer as Soroban.
    The coefficient of x^3 in the expansion of \left( {\sum\limits_{k = 0}^3 {x^k } } \right)\left( {\sum\limits_{k = 0}^2 {x^k } } \right)^2 \left( {1 + x} \right)^3 is 36.

    However that is assuming that the word ‘combos’ means the same thing as multi-set. In the above reply <p,o,p> is a multi-set and is counted only once. If this word ‘combos’ means three letter strings the we would count pop, ppo and opp.

    What is the intended meaning of ‘combos’?
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  5. #5
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    Thanks guys for all your help. I do alright in calculus, but I suck so bad in statistics. I just have to keep trying, I guess.

    And by combos, I mean combinations.
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  6. #6
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    Quote Originally Posted by thefirsthokage View Post
    And by combos, I mean combinations.
    Well strictly speaking, combinations do not involve repetitions.
    Last edited by Plato; January 30th 2007 at 03:26 PM. Reason: spelling
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