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The Banana Theorum?

  1. Apr 17, 2008 #1
    1. The problem statement, all variables and given/known data

    According to my teacher, this is the Banana theorum, but I don't know if this is actually any concrete or just something he coined.

    (I have to prove/derive this)

    Let A be a set with n elements of k different types (such that elements of the same type are regarded as indistinguishable from one another for purposes of ordering.) Let ni. be the number of elements of type i for each integer form 1 to k. Then the number of different arrangements of the elements in A will be

    n!/[tex]\Pi[/tex] (ni!)


    There is supposed to be the usual i=1 below the PI and a k above it.


    2. Relevant equations

    P(k,n) = n!/(n-k)!



    3. The attempt at a solution


    Well, this looks like a like a permutation to me, so i figure it can be derived the same way the equation above can be (I know how to derive that one.) However, since I am fairly green when it comes to the product notation of the denominater, I find myself a little confused on how exactly I can derive this one (and even what this equation is saying.)
     
  2. jcsd
  3. Apr 18, 2008 #2
    The big fat pi is the product analogue to the big fat sigma for sums. Suppose for example that k=3 and n1= 2, n2=3, n3=5. Then the denominator of the fraction would be (2!)(3!)(5!).
     
  4. Apr 18, 2008 #3
    For your reference:

    [tex]\frac{n!}{\prod_{i=0}^k n_i!}[/tex]
     
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