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Permutations/combinations problem

  1. Aug 1, 2008 #1

    JL3

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    I have a problem about combinations and permutations I am trying to solve. Say we have an n-dimensional vector. Each element of the vector can contain any one of [tex]\lambda=3[/tex] values (-1, 0 or +1). Then the number of possible vectors is simply:

    [tex]\lambda^n[/tex]

    If we place the additional restriction that the vector must contain exactly [tex]k[/tex] non-zeros, then it becomes:

    [tex]p=(\lambda-1)^{k}\times\binom{n}{k}=\frac{n!(\lambda-1)^{k}}{k!(n-k)!}[/tex]

    If we change the restriction so that it must contain at most [tex]k[/tex] non-zeros and at least 1 non-zero, then it becomes:

    [tex]p=\sum_{k'=1}^{k}\left[(\lambda-1)^{k'}\times\binom{n}{k'}\right]=\sum_{k'=1}^{k}\frac{n!(\lambda-1)^{k'}}{k'!(n-k')!}[/tex]

    Are my equations correct? Is there a more compact way of expressing this last equation, to get rid of the summation?
     
  2. jcsd
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