How many different ways can a number N be expressed as a sum of K different positive integers? How many different ways can a number $n \in N$ be expressed as a sum of $k$ different positive numbers? 
 A: As noted by Marko Riedel we can limit the analysis to $Q(n,k)$ = number of partitions of $n$ into $k$ distinct parts, where order is not important.
That is to say: Number of strictly increasing sequences, of $k$ terms, with sum = $n$, or
 (changing parameters notation for better clearness) 
$$
\begin{gathered}
  Q(s,q) = \text{No}\text{.}\,\text{of}\,\text{sol}\text{.}\,\text{of}\left\{ \begin{gathered}
  0 < x_{\,1}  < x_{\,2}  <  \cdots  < x_{\,\,q}  \hfill \\
  \sum\limits_{1 \leqslant \,j\, \leqslant \,q} {x_{\,j}  = s}  \hfill \\ 
\end{gathered}  \right.\quad  =  \hfill \\
   = \text{No}\text{.}\,\text{of}\,\text{sol}\text{.}\,\text{of}\left\{ \begin{gathered}
  0 < y_{\,j} \left( { \leqslant s} \right)\quad \left| {\;1 \leqslant j \leqslant q} \right. \hfill \\
  \sum\limits_{1 \leqslant \,j\, \leqslant \,q} {j\;y_{\,j} }  = s \hfill \\ 
\end{gathered}  \right.\quad  =  \hfill \\
   = \text{No}\text{.}\,\text{of}\,\text{sol}\text{.}\,\text{of}\left\{ \begin{gathered}
  0 \leqslant z_{\,j} \left( { < s} \right)\quad \left| {\;1 \leqslant j \leqslant q} \right. \hfill \\
  \sum\limits_{1 \leqslant \,j\, \leqslant \,q} {j\;z_{\,j} }  = s - \frac{{q\left( {q + 1} \right)}}
{2} \hfill \\ 
\end{gathered}  \right. \hfill \\ 
\end{gathered} \tag{1}
$$
where the 2nd step comes from taking $y_{j}=x_{q-j+1}-x_{q-j} \; | \; x_0=0$.
We define Q to be non null only for positive value of the parameters, and to be $=1$ for the empty word $s=0,\;q=0$.
$$
Q(s < 0,q) = 0\quad Q(s,q < 0) = 0\quad Q(0,q) = [q = 0]
$$
(the square brackets indicating the Iverson Bracket)
so that it corresponds to the double generating function
$$
G(x,y) = \sum\limits_{\begin{subarray}{l} 
  0 \leqslant \,q \\ 
  0 \leqslant \,s\, 
\end{subarray}}  {Q(s,q)\;x^{\,s} \;y^{\,q} }  = \prod\limits_{1 \leqslant \,k} {\left( {1 + y\,x^{\,k} } \right)} \tag{2}
$$
From the second relation in (1) it is easy to deduce the recurrence
$$
Q(s,\;q) = \sum\limits_{1 \leqslant \,j\,\,\left( {\, \leqslant \,\,\left\lfloor {s/q} \right\rfloor \,\, \leqslant \,\,s\,} \right)} {Q(s - q\,j,\;q - 1)}  + \left[ {s = 0} \right]\left[ {q = 0} \right]  \tag{3}
$$
A: Number of integer partitions of a number n, denoted by p(n), is much harder to find and no easy formula available.
This wiki page gives various algorithms for finding p(n)
Approximate value for p(n) can be found using Hardy-Ramanujan Asymptotic Partition Formula
$p(n) \approx \dfrac{1}{4n\sqrt{3}}e^\left(\pi \sqrt{\dfrac{2n}{3}}\right)$
This tool generates number of integer partitions online and is very useful
This link is also is related to this.
