I need some help on a problem on combinatorics.


Let $n$ be a natural number greater than $1$ and $k,m$ be two fixed natural numbers not exceeding $n$ with $m\leq\frac{k(k+1)}{2}$.


Let $N=\{{1,2,...,n}\}$ and $S\_i=\{a\_{i1},a\_{i2},...,a\_{ik}\}$ with $S\_i\subseteq{N}$ .


Denote the sum of elements of $S\_i$ by $\sum({S\_i})=a\_{i1}+a\_{i2}+...+a\_{ik}$.


My question is:How many $S\_i\subseteq{N}$ exist with $m\mid\sum{(S\_i)}$?
In other words: how many subsets of $N$ with fixed cardinality $k$ we can choose with the property: the sum of its elements is a multiple of $m$?


(It is not necessary that $S\_i\bigcap{S\_j}=\emptyset$)


Note that the condition $m\leq\frac{k(k+1)}{2}$ is necessary because $S\_i$ could be the set $\{1,2,...,k\}$ with $\sum{(S\_i)}=\frac{k(k+1)}{2}$


I am really wondering if this is a new question and if not, if somebody could help me with this.


Thank you for viewing!



Number of subsets with fixed cardinality k, and sum of elements a multiple of m
It's not clear whether Gaitanas wants his sum to divide $m$ or be a multiple of $m$. If he wants the sum to be a multiple of $m$, then
some related questions are studied in the paper "[Enumeration of Power Sums Modulo a Prime](http://math.mit.edu/~rstan/pubs/pubfiles/35.pdf)" by Andrew M. Odlyzko and Richard P. Stanley, J. Number Theory 10 (1978), 263-272. They study the following question: given a positive integer $n$, an odd prime $p$, and an integer $\alpha$, how many subsets $S$ of $\{1,2,\dots, p-1\}$ are there with 
$$\sum\_{x\in S} x^n \equiv \alpha \pmod p?$$
They also give references to related results, where $p$ need not be a prime.


