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Thread: Combination or Permutation?

  1. #1
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    Combination or Permutation?

    For the problem:
    How many ways are there to re-arrange the letters of LITTLE?
    Would I use combination, permutation or some other method? And how would I do it?
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  2. #2
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    Arrangements with repeated letters

    Hello Lupin
    Quote Originally Posted by Lupin View Post
    For the problem:
    How many ways are there to re-arrange the letters of LITTLE?
    Would I use combination, permutation or some other method? And how would I do it?
    It's basically a permutation problem, but with repeated letters you need to do a bit of extra work. You do it like this:

    If all 6 letters were different - call them $\displaystyle L_1, I, T_1, T_2, L_2, E$ - then there would be $\displaystyle 6!$ arrangements.

    But the $\displaystyle L_1$ and $\displaystyle L_2$ are indistinguishable. And they can be arranged among themselves in $\displaystyle 2! = 2$ ways. So our $\displaystyle 6!$ arrangements will contain each of these $\displaystyle 2!$ arrangements as if they were different. And, of course, they're not. So we need to divide by $\displaystyle 2!$.

    And the same goes for $\displaystyle T_1$ and $\displaystyle T_2$: we need to divide by $\displaystyle 2!$ again, to get rid of the duplication caused by the $\displaystyle T$'s.

    So the final answer is that there are $\displaystyle \frac{6!}{2!2!} = 180$ different arrangements.

    You can generalise this and say that if there are $\displaystyle n$ items, with $\displaystyle a$ items repeated of the first kind, $\displaystyle b$ repeated of the second kind, and so on, the number of different arrangements is:

    $\displaystyle \frac{n!}{a!b!...}$

    Grandad
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