- #1

- 144

- 0

**Hello**

1. Homework Statement

1. Homework Statement

Consider f

_{k}N*→N, k≥0

f

_{k}(n)=Σ j

^{k}n>0

We're looking to establish these identities, using a combinatorial argument

f

_{k}(n)=n if k=0

f

_{k}(n)= 1/(k+1) [ n

^{k+1}+ Σ ( i , k+1) (-1)

^{k+1-i}f

_{i}(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 F

_{j-1}, F

_{j}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)x

^{n-k}y

^{k}

## The Attempt at a Solution

1. |Ω|=n

^{k+1}

2. Ω=∩ Ej for j=1,....,n

3. I just don't get what Fj represents..?

Thanks