# The Banana Theorum?

1. Apr 17, 2008

### Seda

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!/$$\Pi$$ (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. Apr 18, 2008

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!).

3. Apr 18, 2008

### Diffy

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