Counting Subsets of A with k Elements and Sum of r

[soroush1358]

Dear Friends,

I have a question and would be pleased if you help me by suggesting a paper or book to study.

Let A={1,2,...,n}. We consider all the subsets with k elements. How many of these sets have a sum of r ?
e.g.
for n=6, k=3, r=10

{1,3,6}

{1,4,5}

{2,3,5}
Hense the solution in this case is 3.

Thanks for your kind.

 

[Edgardo]

Maybe this helps: http://en.wikipedia.org/wiki/Partition_(number_theory)"

 

[Eynstone]

It's the binomial coefficient (r-1) C (k-1).

 

[ross_tang]

Generating Function of your number

soroush1358, I have answer your question in this post: http://www.voofie.com/content/144/sum-of-subsets-with-k-elements-having-a-total-sum-of-r/" .

You can read more Mathematics related articles in http://www.voofie.com/concept/Mathematics/" .

In short, the generating function of your number is:

$$G_n(x,y) &= (1+x y)(1+x^2 y)(1+x^3 y)...(1+x^n y) \\ &= \prod_{r=1}^n (1+x^r y) \end{align*}$$