# Combinatorial argument probabilities

Hello
1. Homework Statement

Consider fk N*→N, k≥0
fk(n)=Σ jk n>0

We're looking to establish these identities, using a combinatorial argument
fk(n)=n if k=0
fk(n)= 1/(k+1) [ nk+1 + Σ ( i , k+1) (-1)k+1-ifi(n) ] if k>0
the sum is about i from i=0 to i=k-1 and (i, k+1) is the combination.

Consider the experience of k+1 drawing with handoff from a box of n balls numbered
Ej : the maximum of the values obtained on the k+1 draw is j , j=1,......,n
Fj : The drawing are made on {1,.....,j} j=1,....,n
Fo= 1. Find |Ω|
2. Write Ω as an operation on sets E1,....,En
3. Write Ej as an operation on sets Fj-1, Fj j=1....n
4. Find |Fj| j=0....n
5. Find |Ej| j=1....n
6. Find |Ω| through the calculation of |Ej| j=1...n and using the binomial theorem

## Homework Equations

(x+y)n=∑(k, n)xn-kyk

## The Attempt at a Solution

1. |Ω|=nk+1
2. Ω=∩ Ej for j=1,....,n
3. I just don't get what Fj represents..? Thanks

## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?