Proving the Banana Theorum with Permutations

[Seda]

Homework Statement

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!/$$\Pi$$ (ni!)


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

Homework Equations

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

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.)

 

[johnshade]

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!).

 

[Diffy]

For your reference:

$$\frac{n!}{\prod_{i=0}^k n_i!}$$