Selection of $r$ objects out of a set of $N$ objects which are not all distinct I want to know whether or not we can obtain any direct formula for the selection of $r$ objects out a set $\alpha$ that consists of $N$ objects. Where $N$ = $N_1+N_2+N_3+...+N_k$ and $\alpha$ has $N_1$ objects of one kind (all identical), $N_2$ objects of the second kind and so on... I know how to solve such questions through some procedure but I want to know whether or not that procedure has been converted to a direct formula.
 A: I know of two methods for doing this sort of thing in general.
One method is based on the formula $\binom{r+k-1}{k-1}$ to count the number of
sets of $r$ elements of $k$ different types,
but then applying the principle of inclusion-exclusion in order to
eliminate sets that contain more elements of type $i$ than the
original multiset contains (that is, where $x_i > N_i$ for some $i$).
(I am omitting details of how to do this because to do it completely generally would take far too much work; most questions along these lines
that we see on MSE have particular values of $r$, $k$, and $N_i$ for each $i$, which make it possible to apply a specific pattern of inclusion and exclusion.)
The second method uses generating functions. You would be looking 
for the coefficient of $x^r$ in the polynomial
$$
(1 + x + x^2 + \cdots + x^{N_1})
(1 + x + x^2 + \cdots + x^{N_2})\cdots
(1 + x + x^2 + \cdots + x^{N_k}).
$$
For $x \neq 1$ this polynomial is equivalent to
$$
\frac{(1 - x^{N_1 + 1})(1 - x^{N_2 + 1})\cdots(1 - x^{N_k + 1})}{(1-x)^k}.
$$
The value you want is the coefficient of $x^r$ in the
Taylor series of this function about $0$, which you can find by taking the
$r$th derivative of the function at $0$ and dividing by $r!$. 
(Or perhaps better, have your favorite
symbolic math software find the Taylor series for you and read off
the coefficient of $x^r$ in that result.)
